%  FILE: book.toc

\contentsline {chapter}{\numberline {1}Introduction}{1}
\contentsline {chapter}{\numberline {2}Large sets of zeros: conjectures and statistics}{9}
\contentsline {section}{\numberline {2.1}Notation and definitions}{9}
\contentsline {section}{\numberline {2.2}Validity of the RH and correctness of the computational results}{13}
\contentsline {section}{\numberline {2.3}Eigenvalues of random matrices and zeros}{14}
\contentsline {section}{\numberline {2.4}General distribution of gaps between zeros}{20}
\contentsline {section}{\numberline {2.5}Values of $S(t)$}{24}
\contentsline {section}{\numberline {2.6}Extreme gaps between zeros}{30}
\contentsline {section}{\numberline {2.7}Long and short range correlations between zeros}{34}
\contentsline {section}{\numberline {2.8}Lehmer phenomenon}{40}
\contentsline {section}{\numberline {2.9}Large values of $\zeta (1/2+it)$}{43}
\contentsline {section}{\numberline {2.10}Moments of $\zeta (1/2 + it)$}{46}
\contentsline {section}{\numberline {2.11}Distribution of values of $\zeta ( 1/2 + it)$}{49}
\contentsline {section}{\numberline {2.12}Values of $\zeta ' (1/2 + i \gamma )$}{53}
\contentsline {section}{\numberline {2.13}Gram points and blocks}{57}
\contentsline {section}{\numberline {2.14}Violations of Rosser's rule}{61}
\contentsline {chapter}{\numberline {3}Special points for the zeta function}{65}
\contentsline {section}{\numberline {3.1}Introduction}{65}
\contentsline {section}{\numberline {3.2}Computational results}{66}
\contentsline {section}{\numberline {3.3}Diophantine approximation algorithms and special points}{69}
\contentsline {section}{\numberline {3.4}Possible extensions}{71}
\contentsline {chapter}{\numberline {4}Algorithms and their implementation}{75}
\contentsline {section}{\numberline {4.1}Introduction}{75}
\contentsline {section}{\numberline {4.2}Zero-locating program}{77}
\contentsline {section}{\numberline {4.3}Odlyzko-Sch\"{o}nhage algorithm}{80}
\contentsline {section}{\numberline {4.4}Band-limited function interpolation}{88}
\contentsline {section}{\numberline {4.5}Space and time requirements}{93}
\contentsline {section}{\numberline {4.6}Correctness of computational results}{95}
\contentsline {section}{\numberline {4.7}Possible improvements}{106}
\contentsline {subsection}{\numberline {4.7.1}Faster and more accurate computations}{107}
\contentsline {subsection}{\numberline {4.7.2}Greengard-Rokhlin algorithm}{112}
\contentsline {subsection}{\numberline {4.7.3}Computations of low zeros}{116}

%  FILE: preface.tex

\thispagestyle{empty}
\noindent{\Huge {\bf Preface}}

\vspace{.4in}

This monograph presents the results of a computation of over 175 million
consecutive zeros of the Riemann zeta function near zero number $10^{20}$, as well
as of several other large sets of high zeros,
including some near zero number $2 \times 10^{20}$.
These zeros lie about $10^8$ times higher than previously
calculated large sets of zeros, and 
their computation was made possible by 
a fast
algorithm invented by A. Sch\"{o}nhage and the author.
Although the present implementation of this algorithm
is
not
entirely rigorous
due to incomplete control of roundoff errors,
it appears to be highly accurate as well as fast,
and the results indicate that
all the computed zeros satisfy the Riemann Hypothesis.
Various statistical studies of these zeros are presented.
Some of
them
provide numerical evidence about conjectures that go even beyond the Riemann Hypothesis,
and relate the distribution of zeros of the zeta function
to that of eigenvalues of random matrices studied
extensively in physics.
Other studies compare the observed behavior of the zeta function
to known asymptotic estimates.
The computations described in this book were carried
out
on a Cray X-MP supercomputer.

%  FILE: ch1.tex

\chapter{Introduction}\label{sc1}
\hspace*{\parindent}
The $10^{20}$-th zero of the Riemann zeta function equals
$$1/2 + i~ 1
5202440115920747268.6290299 \dd~ .$$
It and
a few of its nearest neighbors are shown in Table~1.1.
All told, almost 176 million zeros near the $10^{20}$-th zero were computed,
as wee as over 101 million zeros near zero number $2 \times 10^{20}$.
These zeros lie almost $10^8$ times higher than any other large
sets of zeros that had been computed before.
This monograph
reports statistics of these and some other high zeros
and describes the algorithms that made these calculations possible

The Riemann Hypothesis (RH) has been subjected to a series of numerical
investigations, starting with unpublished ones by Riemann.
(See \cite{Ed,Od3}
for a history of these computations.)
The latest result is that the RH is true for the first $1.5 \times 10^9$ zeros
(i.e., all zeros up to height $\preceq  5 \times 10^8$)
\cite{LRW2}.
This computation required about 1500 hours on modern
supercomputers (primarily the Cyber~205).
It only separated the zeros, and did not produce accurate
values for them.
The reason for not obtaining values of zeros was that the investigations in this case were very concerned with establishing the validity of the RH, and for that purpose,
as was explained earlier, it is only necessary to separate the zeros of $Z(t)$.
Several other
large sets of zeros (four sets of roughly $10^5$ zeros in each case,
starting with zeros number $10^{10}$, $10^{11}$, $2 \times 10^{11}$,
and $10^{12}$)
have also been computed accurately \cite{Od2}.
Those computations took several tens of hours on Cray-1 and Cray~X-MP supercomputers, and produced values of the zeros that are accurate
to within about $10^{-8}$.
(The $10^{12}$-th zero equals $1/2 + i \gamma$, with $\gamma \approx 2.7 \times 10^{11}$.)
The purpose of those calculations of zeros was partially to check
the validity of the RH,
but the primary goal (and the reason for obtaining
accurate values of the zeros) was to
obtain data about the
distribution of spacings between zeros of the zeta function
so as to compare them to some recent conjectures.
These conjectures, which are described briefly in Section~\ref{sc2} and which go substantially beyond the RH,
originate in the Montgomery pair correlation conjecture, and relate the
behavior of the zeta function zeros to eigenvalues of random
hermitian
matrices that are used to model energy levels in 
many-particle systems in physics, and to the quantum chaos theories.
Agreement between these conjectures and computed values of zeros might
be taken as providing some support for the Hilbert and P\'{o}lya
conjecture that the RH is true because
the zeros of the zeta function correspond to eigenvalues of some positive
operator.
While all these conjectures are highly
speculative, it seemed worthwhile to test them numerically.
As it turned out, the agreement between conjectures and empirical data was excellent in most cases.
A few features of the data that were initially unexpected were in the end
explained by relating the behavior of the zeros to that of the primes.
However, there were some features of the data that were slightly
counterintuitive (such as a slight excess of
small spacings between consecutive zeros over that predicted by the random matrix theories), and so it seemed desirable to obtain data about even higher
zeros of the zeta function.

All large-scale computations of zeros of the zeta function in the last fifty-odd years (as well as Riemann's own unpublished computations \cite{Ed,Od3,Sie1}) relied on the Riemann-Siegel formula \cite{Ed,Sie1,Gab},
which requires
roughly $t^{1/2}$
operations to compute
$\zeta ( 1/2 + it )$ for $t$ a large positive real number.
Recently a much more efficient algorithm for computing the zeta function
was invented by
A. Sch\"{o}nhage and the author
\cite{Od1,OS}.
It enables one to compute all the approximately $T^{1/2}$
zeros of $\zeta ( 1/2 + it)$ in an interval
$T \le t \le T + T^{1/2}$ in about $T^{1/2}$ steps.
(This algorithm is described in detail in Section~\ref{sc4}.
At this point we only note that the above description assumes that
the RH is satisfied by the zeros between heights $T$ and $T + T^{1/2}$, and
that in addition these
zeros are simple and well separated.
All of these conditions are satisfied in all the ranges that have been investigated.)
The new algorithm has now been implemented, and used to compute the zeros described in this paper.
It turns out to be fast in practice as well as in theory,
and for computing large sets of zeros around the $10^{20}$-th zero was at
least $10^5$ times faster than the straightforward application of the Riemann-Siegel formula.
The computations described in this book took about 2000 hours of (otherwise idle) time on a
Cray~X-MP supercomputer, so they were substantial.
However, without the new algorithm they would have been totally infeasible.

The computations that have been carried out with the
new algorithm of \cite{OS}, and that form the basis for this paper, had several goals.
The first was to test the RH numerically.
If the RH is false, then
counterexamples
are probably more likely to be found
at large heights than closer to the origin, since the behavior of the zeta function
is very constrained at low heights.
As it turned out, no counterexample was found.
Another, more important goal, was to extend the numerical studies of \cite{Od2} by
computing accurate values of large sets of high zeros to provide
additional numerical checks on various conjectures about the zeros,
especially about the frequency of occurrence of small gaps.
If the slightly excessive frequency of small gaps that was observed in \cite{Od2}
were to occur again at greater heights, that would cast doubt on many of the
conjectures that have been made.
The latest computations show excellent agreement with these
conjectures
in almost all the measured statistics.
The
excess of small spacings found in \cite{Od2} is still somewhat
ambiguous, though, as will be described in Section~\ref{sc26}.

Another reason for the computations
of this monograph was to produce
various statistics about the zeta function at large heights.
One advantage of the algorithm of \cite{OS} is that once the main step of the computation
is done, it is easy to compute individual values of the zeta function in the covered ranges, and collect many statistics.
Such statistics can be used to test various conjectures (about mean values of the zeta function, for example) and to judge how fast the zeta function approaches its asymptotic behavior.
Some of the statistics presented in Section~\ref{sc2} show amazingly fast convergence to the
asymptotic behavior, while others are far from it.
It is remarkable that
the most noticeable difference between observed and
expected behavior occurs in the study of the distribution of
values of $\log | \zeta (1/2 + it) | $, which in the ranges
that have been investigated is rather far from the normal
distribution that has been rigorously proved to hold
asymptotically (see Section~\ref{sc211} and Figure~2.11.1).
On the other hand, many of the unproved conjectures 
are supported by the numerical data to a surprising degree
(see, for example, Figures~2.4.5 and 2.12.1).

The main conclusion that can be drawn from the data in this paper
is that in many respects the zeta function reaches its asymptotic behavior
slowly, so that even the neighborhood of the $10^{20}$-th zero does not
represent
what happens much higher.
That this slow convergence is observed is not too surprising.
For example, one important question (see Section~\ref{sc29}) concerns the maximal size of the zeta function on the critical line.
It is known that
$$| \zeta (1/2 + it ) | = O(t^\alpha ) $$
for constants $\alpha$ that are a bit less than 1/6, while on the RH one would have
$$| \zeta ( 1/2 + it ) | \le t^{o(1)} ~~\mbox{as}~~  t \to \infty ~.$$
(This is the Lindel\"{o}f conjecture.)
It would be desirable to produce convincing numerical evidence about 
the precise maximal size of $ \zeta ( 1/2 + it ) $.
However, this is hard to do.
The main difficulty is that
near the $10^{20}$-th zero, one has $t \approx 1.5 \times 10^{19}$,
so that $t^{1/6} \approx 1570$, while $( \log t)^2 \approx 1950$,
so it is even hard to distinguish between these two functions that have
entirely different rates of growth.
(Throughout the paper, $\log x$ denotes the natural logarithm of $x$.)

Some of the data from the present computations might also be useful
in other number theoretic investigations.
For example, the
Stark method \cite{St}
for obtaining
lower
bounds for imaginary quadratic number fields with small class numbers depends on
knowledge of pairs of zeros of the zeta function that are very close together.
(Another method for bounding class numbers, that of Montgomery and Weinberger \cite{MW}, depends on zeros of Dirichlet $L$-functions.)

The final
reason for the computations of this paper was to prove
that the new algorithm of \cite{OS} is of practical use, and not just a
theoretical curiosity.
Since this algorithm is complicated, this was not obvious
to start with, and a large section of this paper is devoted to a description
of its implementation, including various modifications that were made to the basic algorithm described in \cite{OS}.
As it turns out, the algorithm is fast, over $ 10^5 $ times
faster than the older algorithms would have been near the
$10^{20}$-th zero.
Moreover,
work on this implementation has suggested many additional modifications, described
in Section~\ref{sc47}, which can probably speed up the algorithm by
another order of magnitude.

The main sets of zeros
that were computed
are listed in Table~1.2.
The entry for $N=10^{20}$, for example, means that 175,587,726 zeros were
computed, starting with zero number $10^{20} - 30, 769, 710$, and ending with zero number
$10^{20} +  144, 818, 015$, and that all these zeros are of the form
$1/2 + i \gamma$ with $\gamma  \approx 1.5 \times 10^{19}$.
Throughout the paper, references to the $N= 10^{20}$ data set will denote
these 175,587,726 zeros
or some subset of them
and similarly for the $N=10^{19} , \dd ,$ data sets.

The starting points for the large data sets listed in Table~1.2 were chosen to be near zeros of round order (such as $10^{20}$), to be easy to refer to.
It was thought that as far as the distribution of zeros is concerned, these
intervals would behave like random ones.
One can also
concentrate on investigating the behavior of $\zeta (1/2 + i t)$ near those $t$ where the zeta function might be expected to behave in an unusual
fashion (e.g., where it is large).
Some such special values of $t$ were found, and the computations
that were carried out there are listed in Tables~3.1.1 and 3.1.2.
(A full explanation of the entries in these tables is given in Section~\ref{sc3}.)
These computations produced many values of the zeta function and of gaps between
zeros that are current
the largest ones known.

While the computations that are described in this book did yield values of zeta
function zeros at much greater heights than would be feasible with older methods, they do have one
serious defect, namely that they are not rigorous.
The validity of the values for the zeros that have been computed (and also of the assertion that all these zeros satisfy the RH)
depends on
the assumption that substantial
cancellation among the
roundoff errors takes place.
This is due largely to the extremely large sizes of the numbers being handled,
and not so much to the new algorithm,
and is explained in detail in Section~\ref{sc4}.
At this point we only mention that the values of zeros that have been obtained are believed
to be accurate to within $\pm 10^{-6}$ or even better
for $N= 10^{20}$.
This belief is based partially on the expected cancellation of errors in the computation.
The strongest argument for the validity of the computations, however,
comes from
several large sets of zeros
which
were computed twice, in
entirely different ways.
That
the numbers being computed were the same follows only
from deep mathematical analysis, and is not obvious from the numbers being
processed.
The resulting duplicate
values for the zeros agreed to the expected degree,
and this is a
strong argument in favor of the validity of the
computations.
These issues are discussed in greater detail in Section~\ref{sc46}.

The remainder of this monograph is organized into three sections.
Section~\ref{sc2} recalls the basic definitions and conjectures, and then presents
the statistics of the large sets of zeros given in Table~1.2.
Section~\ref{sc2} is organized into subsections on a variety of topics, such as large values of the zeta function,
large and small gaps between consecutive zeros, and many others.

Section~\ref{sc3} is devoted to the zeros listed in Table~3.1.1.
First the statistics of these zeros and of various
properties of the zeta
function in those ranges are presented.
Then some
simultaneous Diophantine approximation algorithms (based on the
Lov\'{a}sz
lattice basis reduction algorithm \cite{LLL})
are described, as well as the
ways in which they have been used to produce the points of Table~3.1.1 where the zeta function was expected
to behave pathologically,
and where it does indeed exhibit unusual behavior.

Section~\ref{sc4} describes
the algorithms and computations on which the results
of this monograph are based.
First the basic algorithm of
\cite{OS} is briefly surveyed, and then various
modifications
to it are described.
(Some are minor, while others, such as the use of band-limited function
interpolation, are much more substantial.)
A discussion of various additional modifications that can be utilized in the future
is included
(such as the replacement of the crucial rational function evaluation algorithm of \cite{OS} by somewhat similar algorithms that have been proposed in the context of
astrophysical and fluid dynamics simulations \cite{GR1}, or ways to 
obtain more rigorous results).
There is also a large subsection on the accuracy and validity of the computations of this paper.

%  FILE: ch2.tex

\chapter{Large sets of zeros: conjectures and statistics}\label{sc2}
\section{Notation and definitions}\label{sc20}
\hspace*{\parindent}
The trivial zeros of the zeta function are $-2, -4, -6, \dd$.
We will consider only the
{\em nontrivial zeros},
which lie in the critical strip $0 < \, \mbox{Re}\, (s) < 1$, and are
customarily denoted by $\rho$.
Since for every nontrivial zero $\rho$, $\overline{\rho}$ is also a zero, we will consider
only zeros $\rho$ with $\mbox{Im} \, ( \rho ) > 0$.
(There are no nontrivial zeros $\rho$ with $\mbox{Im} \, ( \rho ) = 0$.)
We number these zeros $\rho_1 , \rho_2 , \dd$ (counting each according to its
multiplicity) so that $0 < \mbox{Im} \, ( \rho_1 ) \le \mbox{Im} \, ( \rho_2 ) \le \dd$.
All the zeros
that have been computed so far are simple and lie on the critical line,
and so can be written as $\rho_n = \frac{1}{2} + i \gamma_n$, $\gamma_n \in R^+$, with
$\gamma_1 = 14.134725 \dd$, $\gamma_2 = 21.022039 \dd$,
$\gamma_3 = 25.010857 \dd$, etc.
In many definitions throughout the paper we will be tacitly assuming that the RH
holds, as otherwise those definitions might not make sense.

Let $N(t)$ denote the number of zeros $\rho$ with $0 < \mbox{Im} \, ( \rho ) \le t$
(counted according to their multiplicity).
Then it is known unconditionally \cite[Chapter~9.3]{Tit2} that
\beql{eq201}
N(t) = \frac{t}{2 \pi}  \log  \frac{t}{2 \pi e} +
O ( \log  t) ~~~\mbox{as} ~~~t \to \In ~.
\eeq
Therefore $\gamma_n \sim 2 \pi n / ( \log n)$ as $n \to \In$.
Since the zeros become denser as the height increases,
and the average vertical spacing between zeros at height $t$ is asymptotic to
$2 \pi / ( \log ( t/(2 \pi )))$, we define the normalized spacing between
consecutive
zeros
$1/2 + i \gamma_n$ and $1/2 + i \gamma_{n+1}$ to be
\beql{eq202}
\delta_n = ( \gamma_{n+1} - \gamma_n ) 
\frac{\log ( \gamma_n / ( 2 \pi ))}{2 \pi} ~.
\eeq
(Here we are assuming that both zeros satisfy the RH.)
It then follows from \eqn{eq201} that the $\delta_n$ have mean value 1 in the sense that for any
positive integers $N$ and $M$,
\beql{eq203}
\sum_{n=N+1}^{N+M} ~ \delta_n = M + O ( \log (NM)) ~.
\eeq

For $t$ real and positive (as will be the case throughout the paper) we define
\beql{eq204}
\theta (t) = \arg [ \pi^{-it/2} \Gamma ( 1/4 + it/2 ) ] ~,
\eeq
where the argument is defined by continuous variation of $s$ in $\pi^{-s/2} \Gamma ( s/2 )$, starting at $s=1/2$ and going up vertically.
We also let
\beql{eq205}
Z(t) = \exp (i \theta (t)) \zeta ( 1/2 + it ) ~,
\eeq
so that $| Z(t) | = | \zeta ( 1/2 + it ) | $.
Then it follows from the functional equation of the zeta function that $Z(t)$ is real, and sign changes of $Z(t)$ correspond to zeros of
$\zeta (s)$ on the critical line.
Almost all calculations of the zeta function on the critical line compute
calculate $Z(t)$ and not $\zeta (1/2 + it )$ (cf.~Section~\ref{sc4}).
However, it is easy to derive one from the other.

The function $\theta (t)$ is monotonic increasing for $t \ge 7$.
For $n \ge -1$, we define the
$n$-th
{\em Gram point}
$g_n$ to be the unique solution $> 7$ to
\beql{eq206}
\theta ( g_n ) = n \pi ~.
\eeq
We have $g_{-1} = 9.666 \dd$, $g_0 = 17.845 \dd$, etc.
Gram points are about as dense as the zeros of $\zeta (s)$ (see Section~\ref{sc212} for a detailed discussion),
but are much more regularly distributed.
In graphs, by a
{\em Gram point scale} we will refer to labeling Gram point
$g_n$ by $n$ (or $n-M$ for some fixed $M$
as $n$ varies).
For example, Fig.~2.1.1 shows $Z(t)$ near zero number $10^{20}$.
Figure~2.1.3 shows $Z(t)$ over a somewhat wider range.

We let
\beql{eq207}
S(t)= \pi^{-1} \arg \zeta (1/2 + it ) ~,
\eeq
where the argument is defined by continuous variation of $s$ in $\zeta (s)$, starting at $s=2$, going up vertically to $s=2+it$, and then
horizontally to $s=1/2 + it$.
(This definition assumes that there are no zeros $\rho$ with $\mbox{Im} \, ( \rho ) = t$.)
The function $S(t)$ has jump discontinuities at heights equal to zeros.
We have
\beql{eq208}
N(t) = 1 + \pi^{-1} \theta (t) + S(t) ~,
\eeq
so that \eqn{eq201} is a consequence of the asymptotic expansion of $\theta (t)$ (which follows from Stirling's formula \cite{HMF})
\beql{eq209}
\theta (t) = \frac{1}{2} t \log (t/( 2 \pi e )) - \pi /8 + O ( t^{-1} ) ~~~\mbox{as} ~~~ t \to \In
\eeq
and the bound \cite[Theorem~9.4]{Tit2}
\beql{eq2010}
| S(t) | = O ( \log  t) ~~~\mbox{as} ~~~ t \to \In ~.
\eeq
Since $N(t)$ is an integer, and $\theta (t)$ is smooth, \eqn{eq208} shows that $S(t)$ jumps at zeros and decreases at a very steady rate
between zeros.
Figure~2.1.2 shows $S(t)$ over the same range of values of $t$ as
in Fig.~2.1.1,
near zero number $10^{20}$.
This range represents typical behavior of $S(t)$ at that height.
(For rare behavior of $S(t)$, see Fig.~3.2.3.)
The function $S(t)$ is of crucial importance in understanding the distribution
of zeros, and Sections~\ref{sc24}, \ref{sc212}, and \ref{sc213} are devoted largely to its properties.

In comparing empirical distributions of various functions, such as $S(t)$ and $\delta_n$, to their conjectured distributions, we will rely extensively
on comparing the moments of their distributions.
The method of moments has fallen into some disrepute in statistics because of its many faults, such as lack of robustness.
(For example, a single outlier in the data can have a large effect,
something we will see in our data.)
However, there are some good reasons for using it.
One is that it is easy to apply.
A more substantial one is that for many of the statistics of the zeta function, such as those of $S(t)$, or of $Z(t)$, computation of moments is currently essentially the only
known tool
that can be used to obtain
rigorous results.
In such cases moments provide the most direct way of comparing empirical distributions to theoretical results.

If a sequence of probability measures with distribution functions $F_n (x)$ is such that for every $k \ge 0$, the $k$-th moment
$$
\mu_n (k) = \int ~ x^k dF_n (x)
$$
converges to $\mu (k)$ as $n \to \In$, then there is a limiting measure with distribution
$F(x)$ whose $k$-th moment is $\mu (k)$.
Furthermore, if the $\mu (k)$ determine their measure uniquely,
and this measure has distribution function $F(x)$,
then the $F_n (x)$ converge to $F(x)$ (in the weak star sense)
\cite[pp.~342--353]{Bil}.
The $\mu (k)$ determine $F(x)$ uniquely if they do not grow too fast \cite{Bil},
\cite[pp.~227--228]{Fel},
so that the normal
distribution, for example, is characterized by its moments.
On the other hand, the log-normal distribution (distribution of $\exp ( \eta )$, where $\eta$ is normal) is not determined uniquely by its moments \cite{Bil},\cite{Fel}.

The standard normal distribution has the density function
\beql{eq2011}
f(x) = ( 2 \pi )^{- 1/2} ~e^{-x^2 /2} ~,
\eeq
with
mean 0 and variance 1.
Often
we will be dealing with quantities (such as $S(t)$) whose
known asymptotic distributions are normal, but which have variances on the order of
$\log \log N$
(for zeros near zero number $N$).
Since $\log \log N$ grows very slowly, it is to be expected that the observed data will
have somewhat different variances, as second order terms are likely to be substantial.
(For $N=10^{20}$,
$\log \log N = 3.82976 \dd$,
so even an additive constant of 1 in the estimate of the variance makes
a considerable difference.)
On the other hand, it is not too unreasonable to hope that the shape of the distribution should be close to the expected one.
To carry out such a comparison, we will often use
a {\em scaled and translated empirical distributions}.
If $x_1 , \dd , x_n$ are samples (of $\delta_m$, say, or other quantities) with mean $a$ and variance $v= \sigma^2$ (so that $\sigma$ is the
{\em standard deviation},
or
{\em rms value}),
\begin{eqnarray}
\label{eq2012}
a & = & \frac{1}{n} ~ \sum_{j=1}^n ~ x_j ~, \\
\label{eq2013}
v & = & \frac{1}{n} ~ \sum_{j=1}^n ~ (x_j -a )^2 ~,
\end{eqnarray}
then the scaled and translated values will be
\beql{eq2014}
x_j^{\ast} = (x_j -a ) / \sigma ~.
\eeq
The $x_j^{\ast}$ have mean 0 and variance 1.
The tables will usually list the $k$-th moment of $x_j^{\ast}$ in the $k$-th entry, but there will be entries giving the ordinary mean $a$
and ordinary variance $v$ that will be marked $k=1^{\ast}$ and $k=2^{\ast}$,
respectively.
In a few cases where the mean $a$ is extremely small, we will use $x_j^{\ast} = x_j / \sigma$.
(These cases will be easy to distinguish because the scaled 1-st moment will not be 0.)

Throughout
this paper, numbers that have ``$\dd$'' at the end are truncated to the form that is shown, while those without ``$\dd$'' are rounded, but the rounding is sometimes up and sometimes down.
Thus, for example, $\pi$ could be represented as 3.14159..., as 3.14159, or as 3.14160.
The log function will always refer to the natural logarithm.
References to maximal values of a function $f(x)$ will usually
mean the values of $f(x)$ for which $|f(x)| $ is maximal.

Constants such as $n_0 , n_1 , \dd$, will generally be different in
different sections, but will be the same within a section.
\section{Validity of the RH and correctness of the computational results}\label{sc21}
\hspace*{\parindent}
The main question about the validity of the computations described in this paper
has to do with size and cancellation of roundoff errors.
This issue is discussed in detail in Section~\ref{sc4}.
Even if we assume that roundoff errors are small
(as they seem to be), there remains
some further lack of rigor.
The set of zeros corresponding to $N=10^{12}$, for example,
is claimed to consist of exactly the zeros numbered $10^{12} - 6, 032$ to
$10^{12} + 1, 586, 163$.
Those 1,592,196 values are indeed zeros of the zeta function strictly between Gram points
of orders $10^{12} - 6, 034$ and $10^{12} + 1, 586, 162$,
provided all the computational steps were correct.
Given the degree of regularity in the locations of those zeros, a
theorem of Turing (see \cite[Theorem~3.2]{Br5}
for a modified and corrected version)
allows us to conclude, for example,
that the 21-st through the 1,592,176-th zeros in our set are indeed
zeros numbered $10^{12} - 6, 012$ through $10^{12} + 1, 586, 143$.
However, this theorem does not exclude the possibility that, for example, the interval
between Gram points $10^{12} - 6, 034$ and $10^{12} - 6, 014$ might contain
some additional zeros.
Since such additional zeros would violate either Rosser's rule (see Section~\ref{sc213})
or even the RH, they seem unlikely to exist,
and in any event would not affect most of the statistics to a noticeable extent,
and so
were assumed not to exist.

There are some further cases of nonrigorous computations in this paper.
For example, the conjectured distribution of the $\delta_n$ (see Section~\ref{sc22}) is complicated, and (as was done in \cite{Od2})
was computed using Van~Buren's program \cite{VB}, with some
modifications by S.~P. Lloyd and this author.
This program
uses an involved
combination of variational procedures and special
function expansions, and no rigorous error analysis for it is known,
although
it appears to be very accurate
(cf.~\cite{Od2}).

Other examples of nonrigorous computation are presented by various
piecewise linear approximations and other interpolation schemes used in the
following sections.
They are all thought to produce accurate results, but no proofs are available.
\section{Eigenvalues of random matrices and zeros}\label{sc22}
\hspace*{\parindent}
Over the last few decades, an extensive collection of results about eigenvalues of certain types of random
matrices has been obtained by mathematical physicists.
The aim of these investigations was to obtain insight into the distribution of energy levels in heavy nuclei, and recently their results have been applied to studies of
energy levels in other kinds of many-particle systems.
Some of the references for this field are
\cite{Be1,Be2,Be3,BG,BGS1,BGS2,BFFMPW,Meh,Por}.
Not only are there many beautiful and mathematically rigorous results
in this area, but there is also experimental evidence that these results do describe
the behavior of physical systems \cite{HPB}.
(Because of the difficulty of the experiments, the physical data, which was obtained through a major effort over the span of several decades, is
sparse and of poor quality compared to the data that can be obtained for the zeta function.)

The eigenvalue results that will be of greatest interest to us are those of the Gaussian unitary ensemble (GUE), which together with the Gaussian orthogonal ensemble (GOE) and the Gaussian symplectic ensemble (GSE) has been studied extensively.
The GUE consists of $n \times n$ complex Hermitian matrices of the form
$A = (a_{jk} )$, where $a_{jj} = 2^{1/2} \sigma_{jj}$,
$a_{jk} = \sigma_{jk} + i \eta_{jk}$ for $j < k$, and $a_{jk} = \overline{a}_{kj} = \sigma_{kj} - i \eta_{kj}$ for $j > k$,
where the $\sigma_{jk}$ and $\eta_{jk}$ are
independent standard normal variables.
(The GOE consists of real symmetric matrices defined similarly.)
The eigenvalues of these matrices are real, and it is their asymptotic distribution, as $n \to \In$, that is of interest.
If we denote the eigenvalues by $\lambda_1 \le \lambda_2 \le \cdots \le \lambda_n$, then
we have
the
{\em Wigner semi-circle law}:
if $M(x)$ denotes the expected number of eigenvalues $\le x$, then for all fixed real $x$,
\beql{eq221}
\lim_{n \to \In} n^{-1} M( x \sqrt{n} ) =
\left\{
\begin{array}{cl}
\df{1}{2 \pi} \dis\int_{-2}^x (4-u^2)^{1/2} du ~, & |x| < 2 ~, \\
~~ \\ [-.09in]
0~, & x \le -2 ~, \\
~~ \\ [-.09in]
1~, & x \ge 2 ~.
\end{array}
\right.
\eeq
This distribution law applies to much more general classes of matrices than those of the GUE and
related ensembles.
For the GUE
(and also
for the
GOE and GSE) a further step is
possible in that one can obtain precise information about the distribution
of spacings between consecutive eigenvalues.
The complete distribution of eigenvalues is known, and one can derive many limit laws.
To do that one normalizes the eigenvalues (basically by stretching the distance between consecutive eigenvalues $\lambda < \lambda'$ by a factor of
$(4n - \lambda^2 )^{1/2} / ( 2 \pi ))$
to make the average nearest neighbor spacing equal to 1.
With this normalization, the distribution of eigenvalues looks the same everywhere
(in the limit as $n \to \In$) and one can in principle
determine any desired statistic of the zeros.
(Doing so in practice means evaluating a definite multidimensional integral,
which is often hard, and gives rise to interesting problems.)
For example, if we use $w$ to denote a normalized eigenvalue in the GUE,
then one finds that for any fixed
$0 \le \alpha < \beta < \In$,
\beql{eq222}
\sE ( | \{ w' :~
w < w' , ~~
w' - w \in [ \alpha , \beta ] \} | ) \sim
\int_\alpha^\beta ~ \lt 1 - \lt
\frac{\sin \pi u}{\pi u} \rt^2 \rt du
\eeq
as $n \to \In$, where $\sE (z)$ is the expectation of $z$.
We say that $1 - ( ( \sin \pi u ) / ( \pi u ) )^2$ is the
{\em pair correlation function}
of the GUE.
(The pair correlation functions of the GOE and the GSE are different.)
Equation~\eqn{eq222} shows, for example, that it is rare for GUE eigenvalues to be close together.
If the $w$'s were obtained by choosing $n$ points independently
and uniformly from the interval $[0, n]$ and letting $n \to \In$,
the pair correlation function would be identically 1.
The GUE pair correlation function in the range $0 \le u \le 3$ is drawn as the solid curve in Fig.~2.4.1,
and is far from being a constant.

If $w$ is a normalized eigenvalue of the GUE, we let $w^{(k)}$ denote the $k$-th
smallest normalized eigenvalue of those that are $> w$.
Then it is known that the $k$-th nearest spacings $w^{(k)} - w$ satisfy
a distribution law;
for all $0 \le \alpha < \beta < \In$,
\beql{eq223}
Prob ( w^{(k)} - w \in [ \alpha , \beta ] ) \sim
\int_\alpha^\beta  p( k-1, u) du
\eeq
as $n \to \In$.
The probability densities $p(k, u)$ (referred to as $p_2 (k; u)$
in many publications, such as \cite{CM2,Mch,MdC}, where the subscript 2 denotes the GUE)
are complicated functions defined in terms of linear prolate
spheroidal functions.
For methods of computing them, see \cite{MdC,Od2}.
Graphs of $p(0, u)$ and $p(1, u)$ are given by the solid lines
in Figs.~2.4.4 and 2.4.6, respectively.
Those graphs show the ``rigidity'' of the GUE;
the eigenvalues repel each other and most of the time are close to the expected
distance from their neighbors.
For all $u \ge 0$,
\beql{eq224}
1 - \lt \frac{\sin \pi u}{\pi u} \rt^2 = \sum_{k=0}^\In ~
p(k, u ) ~.
\eeq
We note for future reference that the $p(k, u)$ have the following
Taylor series expansions around 0
\cite{Meh,MdC}:
\begin{eqnarray}
\label{eq225}
p(0, u) & = & \frac{\pi^2}{3} u^2 -
\frac{2 \pi^4}{45} u^4 + \frac{\pi^6}{315} u^6 + \cdots ~, \\
\label{eq226}
p(1~, u) & = & \frac{\pi^6}{4050} u^7 + \cdots , \\
p(2, u) & = & \frac{\pi^{12}}{5358150000} u^{14} +... ~.
\nonumber
\end{eqnarray}

The normalized eigenvalues in the GUE have (in the limit as $n \to \In$) a stationary
distribution.
This means that clusters of eigenvalues have the same distribution no matter where in the
spectrum they are located.
However, this distribution is not Markovian, so that the distribution
of an eigenvalue depends not just on the preceding eigenvalue, but on all previous ones as well.

The basic results
about distribution of GUE eigenvalues are completely rigorous.
However, they do have many gaps.
One of them is that the results are obtained by averaging over the full
ensemble of GUE matrices.
It is conjectured that if one considers a large random GUE matrix,
the distribution of its eigenvalues will be close to that of the
entire ensemble
with high probability.
Although numerical calculations confirm this conjecture, there is no proof of it.
Also, it is thought that entries of the matrix do not have to be of exactly
the form specified above for the GUE result to hold.

The main goal of this monograph (and of the preceding paper \cite{Od2}) is to test the conjecture,
which will be referred to as the
{\em GUE hypothesis},
the {\em GUE theory}, or simply the {\em GUE}, that the zeros of the zeta function
behave like eigenvalues of the GUE.
More precisely, it is conjectured that the $\delta_n$ behave asymptotically like
$w^{(1)} - w$ in the GUE, so that for any $0 \le \alpha < \beta < \In$,
\beql{eq227}
M^{-1} | \{ n :~
N+1 \le n \le N+M ,~~\delta_n \in [ \alpha , \beta ] \} | \sim
\int_\alpha^\beta p(0, u) du
\eeq
as $M, N \to \In$ with $M$ not too small compared with $N$.
Similarly, it is conjectured that
\beql{eq228}
M^{-1} | \{ n :~ N+1 \le n \le N+M ,~~\delta_n + \delta_{n+1} \in 
[ \alpha , \beta ] \} | \sim \int_\alpha^\beta  p(1, u) du ~.
\eeq
More generally, the same reasoning leads one to expect that for any fixed $k$,
the empirical distribution function of $\delta_n , \delta_{n+1} , \dd , \delta_{n+k}$ for $N+1 \le n \le N+M$ approaches the stationary
process that holds for the GUE.

If the GUE
hypothesis
is true, it might be
interpreted as providing some support for the Hilbert and
P\'{o}lya
conjectures \cite{Be2,Be3,Mon1,Od3} which predict that the RH is true because
the zeros of the zeta function correspond to eigenvalues of a positive linear operator.
The argument is that if such an operator exists, its eigenvalues might be similar
to those of a random operator (especially if, as is conjectured for the GUE, most random
operators have similar eigenvalue distributions), and a random linear
operator ought to be the limit of a sequence of random matrices.

If the GUE hypothesis were true, that would also be of
interest in physics, as the zeta function could then be used as a
model of quantum chaos \cite{Be2,Be3}.

The main theoretical support and inspiration for the GUE hypothesis comes from
H. Montgomery's work on the pair correlation function of the zeros of the zeta function.
Under the assumption of the RH, Montgomery showed \cite{Mon1,Mon2} that if we define
\beql{eq229}
F( \alpha , T) = 2 \pi (T \log T)^{-1} \sum_{0 < \gamma \le T \atop 0 < \gamma' \le T}
T^{i \alpha ( \gamma - \gamma' )} 
\frac{4}{4+( \gamma - \gamma' )^2}
\eeq
for $\alpha$ and $T$ real, $T \ge 2$, then
\beql{eq2210}
F( \alpha , T) = (1+ o(1)) T^{- 2 \alpha} \log  T + \alpha + o(1) ~~~\mbox{as}~~~
T \to \In ~,
\eeq
uniformly for $0 \le \alpha \le 1$.
Montgomery also observed that if the primes are distributed sufficiently uniformly
in arithmetic progressions, then
\beql{eq2211}
F( \alpha , T ) =1 + o(1) ~~~~\mbox{as} ~~~~T \to \In
\eeq
uniformly for $\alpha \in [a, b]$, where $1 \le a < b < \In$
are any constants.
If the conjecture \eqn{eq2211}
were true, then one would find that
for any $0 < \alpha < \beta < \In$,
\beql{eq2212}
\begin{array}{c}
N^{-1} | \{ (n, k) :~
1 \le n \le N ,~ k \ge 0 ,~~
\delta_n + \delta_{n+1} + \cdots +
\delta_{n+k} \in [ \alpha , \beta ] \} | \\
~~ \\
\sim \dis\int_\alpha^\beta \lt 1 - \lt
\df{\sin \pi u}{\pi u} \rt^2 \rt du
\end{array}
\eeq
as $N \to \In$.
The relation \eqn{eq2212}
is known as the Montgomery pair correlation conjecture.
It says that the pair correlation of the zeros of the zeta function is the same as that of the GUE.
Since the pair correlations of the GOE and GSE are different,
(indeed, they are even inconsistent with \eqn{eq2210}),
this leads one to expect that the zeros might behave
like eigenvalues of the GUE rather than GOE or GSE.
Therefore the discussion above was concentrated on
the GUE distribution.
(One possible implication of this observation is that the hypothetical Hilbert-P\'{o}lya operator
is likely to be complex.)

Montgomery's hypothetical result \eqn{eq2210} and the conjectures \eqn{eq2211}
and
\linebreak
\eqn{eq2212} are the main theoretical evidence we have in favor of the GUE hypothesis,
and the two conjectures depend on far-reaching assumptions about pseudorandom
behavior of primes.
Some further evidence in favor of the GUE
hypothesis was provided by Ozluk \cite{Oz1}, who showed that if one considers a function
similar to $F( \alpha , T)$, but where one sums over zeros of many
Dirichlet $L$-functions, then under the assumption of the Generalized Riemann
Hypothesis for these $L$-functions, the analog of Montgomery's conjecture \eqn{eq2211}
is true for $1 \le \alpha \le 2$.
Some further slight support for the GUE hypothesis is provided by new results of
Ozluk \cite{Oz2} on zeros of Dirichlet $L$-functions close to the real axis.

Extensive numerical evidence in favor of the GUE hypothesis was presented in \cite{Od2}.
It was based largely on computed values of $\gamma_n$,
with
$1 \le n \le 10^5$ and
$10^{12} +1 \le n \le 10^{12} + 10^5$.
With some slight exceptions (such as the slight excess of small $\delta_n$
that was mentioned in the Introduction)
this evidence was in excellent agreement with the GUE hypothesis, and the degree
of agreement improved dramatically as one went from the first $10^5$ zeros
to those near zero number $10^{12}$.
Some numerical evidence for the pair correlation conjecture for Dirichlet $L$-functions has been obtained
since then by Hejhal \cite{Hej5}.
It involved computations of a few quadratic character $L$-functions for several
moduli at large heights.
Much more extensive data have been obtained by Rumely \cite{Rum}, who computed all
zeros of all $L$-functions to small moduli up to height 2500.
His evidence also supports the GUE hypothesis.

Various theoretical results and conjectures related to the GUE theories and the pair correlation conjecture have been obtained in recent years.
Some of the references are \cite{Be2,Be3,Be4,Fu8,Gal2,Gal3,Gal4,GM,Go1,Go2,Go3,GG,GHB,GM,HB1,Mue2}.
\section{General distribution of gaps between zeros}\label{sc23}
\hspace*{\parindent}
Figure~2.4.1 shows how well the pair correlation conjecture is satisfied.
The solid line is the GUE prediction $y = 1- (( \sin \pi x ) / ( \pi x ))^2$.
The scatterplot is based on about $8 \times 10^6$ zeros
near zero number $10^{20}$.
Let
$$
\begin{array}{l@{~}l@{~}l}
n_1 & = & 10^{20} - 15, 409, 240, \\
n_2 & = & 10^{20} - 13, 366, 460, \\
n_3 & = & 10^{20} - 10, 302, 282, \\
n_4 & = & 10^{20} - 6, 216, 711, \\
n_5 & = & 10^{20} - 42, 778, \\
n_6 & = & 10^{20} + 15, 316, 087, \\
n_7 & = & 10^{20} + 46, 073, 204, \\
n_8 & = & 10^{20} + 47, 098, 588,
\end{array}
$$
and
$$V = \{ n :~ n_i \le n < n_i + 10^6 ~~~\mbox{for some} ~~i, ~~~
1 \le i \le 8 \} ~.
$$
Then for each interval $I = [ \alpha , \beta )$ with $\alpha = k/20$,
$\beta = \alpha + 1/20$, $0 \le k < 60$, a star is placed at the point
$x = ( \alpha + \beta ) /2$, $y = a_{\alpha , \beta}$, where
\beql{eq231}
a_{\alpha , \beta} =
\frac{20}{8 \times 10^6} \left |
 \{ (n, k) :~ n \in V ,~~k \ge 0 , ~~ \delta_n + \cdots + \delta_{n+k} \in 
[ \alpha , \beta ) \} \right | ~.
\eeq
The solid line is the GUE prediction
$y=1-(( \sin \pi x ) / ( \pi x ))^2$.
As can be seen, the agreement between the conjectured and observed values is excellent.

Figure~2.4.2 presents similar data, but this time based on just $10^6$ values of
$n$;
$n_9 \le n < n_9 + 10^6$, $n_9 = 10^{12} - 6, 032$.
A comparison of these two graphs with Figures~1 and 2 of \cite{Od2} is instructive.
Those figures show similar graphs, but based in each case on $10^5$ zeros starting with zeros number 1 and $10^{12} +1$.
The scatterplot of Fig.~2.4.2 is much smoother than that of Fig.~2 of \cite{Od2},
because the former is based on $10^6$ instead of $10^5$ samples, and so the sampling error is smaller.
That same reason explains why the scatterplot of Fig.~2.4.1 looks smoother than that of Fig.~2.4.2.
Even if we make allowances for the different sample sizes, though, it is clear that the agreement between empirical and predicted values improves dramatically from $N=1$ to $N= 10^{12}$, and improves
even more between $N=10^{12}$ and
$N=10^{20}$.
In all cases, the empirical data has more pronounced peaks and troughs than expected, but this effect decreases as the height increases.

Some of the pair correlation function oscillations can be seen even for normalized
spacings that exceed 3.
Figure~2.4.3 shows a graph based, just like Fig.~2.4.1, on $8 \times 10^6$ zeros near zero
number $10^{20}$.
Here, though, the scatterplot was smoothed slightly by applying the lowess
function of \cite{BC}
(an implementation of Cleveland's robust locally weighted regression \cite{Clev}).
The reason for this smoothing is that even with $8 \times 10^6$ zeros,
each of the $a_{\alpha , \beta}$ defined in \eqn{eq231} corresponds to about
$4 \times 10^5$ counts $(n, k)$.
Therefore we can expect random sampling errors about $(4 \times 10^5 )^{1/2}$, which gives a variation of about $1.6 \times 10^{-3}$ in the value of $a_{\alpha , \beta}$.
Given the small variation in the GUE prediction $y=1- (( \sin \pi x ) /( \pi x ))^2$ over the range $3 \le x \le 5$, this random sampling error
produces a confusing picture if the data is not smoothed.
(Another, but slightly less effective way to produce a better picture is to use sampling
intervals larger than 1/20.
The resulting picture is similar to that of Fig.~2.4.3.)

Figure~2.4.3 shows that the empirical pair correlation function, even for $N=10^{20}$, has peaks and troughs that are more pronounced than those of the conjectured distribution,
at least in the range $3 < x < 5$.
This is also true in the range $5 < x < 10$.

Figures~2.4.4 and 2.4.5 show the distribution of the normalized spacings $\delta_n$ for $N=10^{12}$ and $N=10^{20}$,
based on the 1,592,196 and 78,893,234 zeros,
respectively, that have been computed.
Thus, for example, in Fig.~2.4.4 a star is plotted at $x= ( \alpha + \beta ) /2$,
$y= b_{\alpha , \beta }$ for $\alpha = k/20$,
$\beta = \alpha + 1/20$, $0 \le k \le 59$, where
\beql{eq232}
b_{\alpha , \beta} =
\frac{20}{1592195} \left | 
 \{ n :~ 10^{12} - 6032 \le n \le 10^{12} + 1586162,~~ \delta_n \in [ \alpha , \beta ) \} \right | ~.
\eeq
The solid lines are the GUE predictions, $y= p(0, x)$.
Similarly, Figures~2.4.6 and 2.4.7 show the distribution of $\delta_n + \delta_{n+1}$.
(Similar graphs based on the first $10^5$ zeros are contained in \cite{Od2}.)

The graphs show excellent agreement between conjecture and numerical data,
and, as was to be expected, the degree of agreement increases substantially as one goes from $N=1$ to $N= 10^{12}$, and then improves a bit more as one goes to $N=10^{20}$.
That the disagreement is greater for $\delta_n + \delta_{n+1}$ than for $\delta_n$ is to be expected,
given that $S(t)$ is small.
(See Section~\ref{sc24} for a discussion of this.)

A quantitative measure of the agreement between observed and conjectured
distributions is shown in Tables~2.4.1 through 2.4.3, which display moments of
distributions.
For each set of $M$ zeros, $K < n \le K+M$
$(M=1, 592, 196$ for $N=10^{12}$, 78,893,234 for $N=10^{20}$, etc., where $K$ is close to $N$.)
Table~2.4.1 displays
\beql{eq233}
(M-1)^{-1} \sum_{n=K+1}^{K+M-1} ( \delta_n -1 )^k ~,
\eeq
while Table~2.4.2 shows
\beql{eq234}
(M-2)^{-1} \sum_{n=K+1}^{K+M-2}  ( \delta_n + \delta_{n+1} -2)^k ~,
\eeq
in each case for $2 \le k \le 10$.
(The values for $N=1$ are taken from \cite{Od2}.)
Table~2.4.3 shows moments of $\log  \delta_n$, $\delta_n^{-1}$,
and $\delta_n^{-2}$.
The values predicted by the GUE are also shown.

Tables~2.4.1 to 2.4.3 show satisfactory agreement between observed values
and conjectured ones, with the degree of agreement increasing as the height of the
zeros increases.
(The slightly anomalous value for the moment of $\delta_n^{-2}$ for $N=10^{18}$ is due to one extremely small $\delta_n$ that is very unusual and will be
discussed in Sections~\ref{sc25} and \ref{sc27}.)

The Kolmogorov test \cite[Section~30.49]{KS}
yields a method for measuring the agreement between the observed distribution of the $\delta_n$ and the GUE predictions.
If samples $x_1 , \dd , x_n$ are drawn from a distribution with a continuous
cumulative distribution function $F(z)$, let $F_e (z)$ denote the sample
distribution function:
$$F_e (z) = n^{-1} | \{ k :~ 1 \le k \le n ,~~ x_k \le z \} | ~.$$
The
Kolmogorov
statistic is then
\beql{eq235}
D = \dis\sup_z |F_e (z) - F(z) |~.
\eeq
If the $x_i$ are drawn independently from the distribution
characterized by
$F(z)$, then
\cite[Eq.~30.132]{KS}
\beql{eq236}
\lim_{n \to \In} Prob (D > un^{- 1/2} ) = g(u) ~,
\eeq
where
\beql{eq237}
g(u) = 2 \sum_{r=1}^\In (-1)^{r-1} \exp (-2r^2 u^2 ) ~.
\eeq
Table~2.4.4 gives the
Kolmogorov
statistic $D$ for $\delta_n$ and $\delta_n + \delta_{n+1}$ for several
blocks of $10^6$ consecutive values of $n$.
The set denoted by $N=10^{12}$ corresponds to
$n_9 \le n < n_9 + 10^6$;
the ones denoted by $N=10^{20} (a)$,
$N=10^{20} (b)$, and $N= 10^{20} (c)$ start at
$n=n_6$, $n=n_8$ and $n=n_5$,
respectively,
where the $n_i$ were defined at the beginning of this section.
The ``$N=10^{12}$ vs. GUE'' entry, for example, gives the
Kolmogorov
statistic
of the $N=10^{12}$ set when it is compared to the GUE distribution.
For each value of $D$, the ``prob.'' column gives an estimate that this statistic
would arise if the $\delta_n$ ($\delta_n + \delta_{n+1}$, respectively) were drawn independently for each $n$ from the GUE
distribution.
This estimate is obtained by evaluating $g( D \times 1000)$.
The ``$N=10^{20} (a)$ vs. $N=10^{20} (b)$'' row of the table was
obtained by constructing a continuous distribution from the $N=10^{20} (b)$ data and computing
the
Kolmogorov
statistic
for the discrete $N=10^{20} (a)$ data against this continuous distribution.

What is apparent from Table~2.4.4 is that as the height increases,
the empirical distributions of $\delta_n$ and $\delta_n + \delta_{n+1}$ do approach that of the GUE.
When one computes the $D$ statistic for the $\delta_n$ in the 10 blocks of $10^5$ consecutive zeros that are contained in the $N=10^{20} (b)$ set, one obtains
values ranging between 0.002 and 0.0031, which correspond to probabilities between
0.83 to 0.3 of occurring if the $\delta_n$ were drawn from the GUE distribution.
Thus for sets of $10^5$ zeros around zero number $10^{20}$,
it is essentially impossible to distinguish the empirical distribution of the $\delta_n$ from the
expected one.
(For $\delta_n + \delta_{n+1}$, the corresponding $D$ values are 0.0035 and 0.00555, which gives probabilities of 0.17 and 0.004, so the fit here is slightly
worse.)

The comparison of the three different sets of $10^6$ zeros near zero number $10^{20}$ to each other is revealing.
The Kolmogorov
statistics
$D$ are small (especially for $\delta_n$), and indicate that all three sets come from
essentially the same distribution.
What seems to be happening is that at each height, when we examine large sets of zeros, the $\delta_n$ and $\delta_n + \delta_{n+1}$ behave as if they were drawn
independently from some distributions that depend on $t$,
change slowly as $t$ changes, and tend to the GUE distributions as
$t \to \In$.
\section{Values of $S(t)$}\label{sc24}
The upper bound \eqn{eq2010} for $S(t)$ is the best that is known unconditionally.
The Lindel\"{o}f Hypothesis (see Section~\ref{sc28}) implies that
$|S(t)| = o( \log  t)$ as $t \to \In$,
while the RH implies \cite{Tit2} that
\beql{eq241}
|S(t)| = O \lt \frac{\log t}{\log \log t} \rt ~~~\mbox{as} ~~~ t \to \In ~.
\eeq
The true rate of growth is thought to be much smaller.
The best lower bound that has been proved under the RH is that of
Montgomery \cite{Mon3}, who showed
\beql{eq242}
S(t) = \Omega_\pm \lt \lt \frac{\log t}{\log \log t} \rt^{1/2} \rt ~~~\mbox{as} ~~~ t \to \In ~.
\eeq
(The best unconditional bound, due to Tsang \cite{Ts1,Ts2},
replaces the square root in \eqn{eq242} by a cube root.)
Montgomery [Mon3] has conjectured that the quantity on the right side of \eqn{eq242}
represents the correct rate of growth of $S(t)$, and Joyner \cite{Joy2} has presented a
heuristic argument supporting this conjecture.
As we will see in Section~\ref{sc25}, the GUE suggests that $|S(t)| $ might occasionally
get as large as $( \log  t)^{1/2}$, which would contradict the Montgomery conjecture.
In any case, it is thought likely that
\beql{eq243}
|S(t)| \le ( \log  t)^{1/2 + o(1)} ~~~\mbox{as} ~~~ t \to \In ~.
\eeq
Some lower bounds for $S(t+h) - S(t)$ are also known, see \cite{Ts1,Ts2}, for example.

Not only is $S(t)$ small, but its oscillations tend to cancel out.
If we define
\beql{eq244}
S_1 (t) = \int_{t_0}^t  S(u) du ~,
\eeq
then $|S_1 (t) | = O( \log  t)$ unconditionally, and
$|S_1 (t) | = O( ( \log  t) ( \log \log t)^{-2} )$ on the RH \cite{Tit2}.
The true maximal magnitude of $|S_1 (t) | $ is probably again around
$( \log t)^{1/2}$.
(See \cite{Ts1,Ts2} for lower bounds.
The estimate $|S_1 (t) | = o( \log  t)$ is equivalent to the
Lindel\"{o}f Hypothesis,
see Notes to Chapter~13 of \cite{Tit2}.)
Furthermore, if one chooses $t_0$ appropriately,
then one obtains
\beql{eq245}
\int_0^t  S_1 (u) du = O( \log  t) ~~~\mbox{as} ~~~ t \to \In ~.
\eeq
(The same property applies to further iterations of this process.)
In addition,
$$\lim_{T \to \In}  T^{-1} \int_0^T  S_1 (t)^2 dt = c
$$
exists for a constant $c > 0$ (Theorem~14.19 of \cite{Tit2}).

Selberg \cite{Sel2} proved, under the assumption of the RH,
that for every fixed positive integer $k$,
\beql{eq246}
\int_0^T  S(t)^{2k} dt =
\frac{(2k)!}{k! (2 \pi )^2k} T ( \log \log T)^k ( 1 + O( ( \log  \log T)^{-1} ))
\eeq
as $T \to \In$.
Later \cite{Sel3} he proved similar estimates unconditionally, with $( \log  \log  T)^{-1}$ in the remainder
term replaced by $( \log  \log  T)^{-1/2}$.
Although it was apparently not noticed right away, these results imply (unconditionally) that
$S(t)$ is asymptotically normally distributed with mean 0 and variance $2 \pi^2 \log \log t$, so that for
$\alpha < \beta$,
\beql{eq247}
\lim_{T \to \In} T^{-1} \left | 
\left \{ t : 0 \le t \le T , \frac{S(t)}{( 2 \pi^2 \log \log T)^{1/2}}
\in ( \alpha , \beta ) \right \} \right | = (2 \pi )^{-1/2}
\int_\alpha^\beta e^{-x^2 /2} dx ~.
\eeq
For further results on moments and distributions of $S_1 (t)$, $S(t+h) - S(t)$, and related
functions, see
\cite{Fu1,Fu2,Fu3,Fu4,Fu8,GM,Gh1,Gh2,Go2,Joy1,Ts1,Ts2}.
Goldston \cite{Go2} has improved the estimate \eqn{eq246} for $k=1$ by showing, under the assumption of the RH, that
\beql{eq248}
\int_0^T S(t)^2 dt = \frac{T}{2 \pi^2} \log \log T +
\frac{T}{2 \pi^2} \lt c_1 + \int_1^\In
F( \alpha ,T) \alpha^{-2} d \alpha \rt + o(T)
\eeq
as $T \to \In$, where $F( \alpha , T)$ is defined by \eqn{eq229}, and $c_1$ is a constant,
\beql{eq249}
c_1 = c_0 + \sum_{m=2}^\In \sum_{p} \lt - \frac{1}{m} +
\frac{1}{m^2} \rt \frac{1}{p^m} ~,
\eeq
where $c_0 = 0.577 \dd$ is Euler's constant.
(The sign of the $m^{-1}$ term is wrong in \cite{Go2}.)
If Montgomery's pair correlation conjecture \eqn{eq2211} holds, then
\linebreak
$\int_1^\In F( \alpha , T) \alpha^{-2} d \alpha$ is asymptotic to the constant 1, but if his conjecture were to fail,
it is conceivable that the second order term in the asymptotic expansion of $\int_0^T S(t)^2 dt$ might oscillate.

Table~2.5.1 presents data on the moments of $S(t)$.
Statistics were collected on two intervals of the form $( \gamma_n , \gamma_{n+10^6} )$, where
$n=n_1 = 10^{12} - 6, 032$ for the $N=10^{12}$ data,
and $n=n_2 = 10^{20} - 48, 778$ for the $N=10^{20}$ data.
The average values of $S(t)$ and $S(t)^2$ for these sets are given
in the $k=1^{\ast}$ and $k=2^{\ast}$ zeros.
To obtain a good comparison with the asymptotic normal distribution, the other moments were
scaled, so that if we let $\sigma^2$ be the mean value of $S(t)^2$,
then the $k=1, 2, \dd , 8$ entries denote the average values of $( \sigma^{-1} S(t))^k$,
and the $k=|1| , |3| $, and $|5| $ entries the average values of $| \sigma^{-1} S(t) |^k$.
Finally, the last column gives the corresponding values for the standard normal
distribution.
As we can see, the agreement between empirical values and asymptotic ones is reasonably good,
and is somewhat better for $N=10^{20}$ than for $N=10^{12}$.

$S(t)$ has jumps discontinuities by 1 at zeros and decreases monotonically between zeros
with derivative very close to $-1$ (on Gram point scale). Since there is asymptotically
one zero per Gram point, the smallest mean values of $S(t)^{2k}$ for any $k \in Z^+$ that is at all conceivable would be obtained by having a zero exactly halfway
between every two neighboring Gram points.
This would yield a mean value of $S(t)^2$ of 1/6.
The values that are observed, 0.23 for $N=10^{12}$ and 0.26 for $N=10^{20}$,
are not much larger than that.

That the distribution of $S(t)$ is close to the normal one can be seen visually
in Fig.~2.3.1.
This figure is based on determining for what fraction of values of $t \in ( \gamma_{n_2 , \gamma_{n_2} + 10^8} )$ we have $S(t) \in [k/100 , (k+1)/100)$, and then scaling
the resulting histogram by $\sigma$ to produce a graph that can be compared to that of the standard normal distribution.
It is curious that the observed distribution of $S(t)$ is less peaked than the normal one,
whereas in most of the other comparisons the empirical distributions have sharper peaks than expected.
It is especially interesting to compare Fig.~2.5.1 to Fig.~2.11.1,
which compares the distribution of $\log  | Z(t) | $ (up to a constant the harmonic conjugate of $S(t)$) to the normal distribution.
In both cases the limiting distributions are known to be normal (even without assuming the RH), but the observed deviations from normal behavior are different for $S(t)$
and $\log  |Z(t)| $, and are much more pronounced in the latter case.

The area between the two curves in Fig.~2.5.1 is 0.023.
For the corresponding figure using the $N=10^{12}$ data, the area is 0.029.

Since both $S(t)$ and its integral $S_1 (t)$ are small, 
we can expect that $S(t)$ will have many sign changes, 
and several results in this direction have been proved,
the strongest ones being due to Ghosh \cite{GL1} and Mueller \cite{Mue1},
but they are all weak.
For example, Mueller proves that gaps between consecutive zeros of $S(t)$ are $O( \log \log \log t)$.
On the other hand, we know that
$S(t)$ has a limiting normal distribution with variance on the order of $( \log \log t)^{1/2}$ and mean close to 0, and that it cannot vary too widely
(in particular,
except for jump discontinuities at zeros it
is monotone decreasing with derivative
close to
$- \log t$).
Therefore
we might expect that the ratio of the number
of zero crossings of $S(t)$ for $t \in ( \gamma_N , \gamma_{N+M} )$ to $M$ might be
roughly the fraction of $t$ in $ ( \gamma_N, \gamma_{N+M} ) $
for which $|S(t) | \le 1$.
This suggests that on average there ought to be on the order of
$( \log  \log  t)^{-1/2}$ zeros of $S(t)$ per Gram interval.

The number of sign changes of $S(t)$ in the intervals that have been investigated can be
determined easily from the statistics of Gram blocks and of
exceptions to Rosser's rule that have been collected.
When $g_n$ is a good Gram point (see Section~\ref{sc212} for a definition) that is not close to an exception to
Rosser's rule,
and is not a zero of the zeta function, then $S(g_n ) = 0$,
and $S(t)$ changes sign at $g_n$.
We will count this sign change as occurring in the Gram interval $[g_n , g_{n+1} )$.
If $B(n, k)$ is a Gram block that has exactly $k$ zeros, then
an easy accounting shows that $S(t)$ has exactly 2 sign changes in $B(n, k)$.
On the other hand, when $B(n, k)$ is an exception to Rosser's rule
(see Section~\ref{sc213} for definitions),
and $[g_m, g_{m+r} )$ is the smallest union of Gram blocks that contains
both the exception and the excess zeros, then a similar
accounting shows that $[g_m , g_{m+r} )$ contains exactly 2 sign changes.
Thus if Gram's law (see Section~\ref{sc212}) held universally,
we would have an average of 2 sign changes of $S(t)$ for every zero of $zeta (s)$.
Departures from Gram's law lower this average.
Table~2.5.2 shows the computed averages for the different data sets.
There is a steady decrease in the average, but it is slow.
Since the argument in the preceding paragraph suggests a rate of decrease of
$( \log  \log  t)^{-1/2}$, this is not surprising.

For every exception $B(n, k)$ to Rosser's rule
there is a $t$ nearby with
$|S(t)| \ge 2$ (and even $|S(t)| > 2$, if zeros do not coincide with Gram points,
as seems likely).
Statistics about these large values of $S(t)$ were collected during investigation of exceptions to Rosser's rule.
Large values of $S(t)$ are of special interest because it is only when $S(t)$
is large that unusual behavior of the zeta function
can take place.
Locally extreme values of $S(t)$ occur at zeros.
(Each zero has
associated to it two values of $S(t)$,
the limits of $S(t)$ as $t$ approaches the zero from the right or the left.)
Table~2.5.3 shows the values of $S(t)$ for which $|S(t)| $ is largest in absolute
value, as well as the number of zeros at which $|S(t)| > 2.3$ divided by the number of exceptions to Rosser's rule.
The largest value of $|S(t)| $ that was found in these computations is 2.7916, while among the
first $1.5 \times 10^9$ zeros the largest such value is
2.3137 \cite{LRW2}.
(A point $t$ at which $|S(t)| = 2.8747$ was found later in the computations described in Section~\ref{sc3}.)
Earlier computations established that $|S(t)| < 1$ for $7 < t \le 280$, and $|S(t)| < 2$ for $7 < t \le 6.8 \times 10^6$.

The values of $S_1 (t)$ were investigated in the two intervals $( \gamma_n , \gamma_{n+10^6} )$, where
$n = 10^{12} - 6, 032$ (for the $N=10^{12}$ set) and $n = 10^{20} - 48, 778$
(for $N=10^{20}$).
The values of $S_1 ( \gamma_n )$ were
chosen to make

\beql{eq2410}
\int_{\gamma_n}^{\gamma_m}  S_1 (t) dt = 0 
\eeq
for $m=n+ 10^6$.
The data that were obtained are summarized in Table~2.5.4;
the mean of $S_1 (t)^4$, for example, refers to
$$\frac{1}{\gamma_m - \gamma_n}  \int_{\gamma_n}^{\gamma_m}  S_1 (t)^4 dt ~.
$$
In addition to the uncertain choice of $S_1 ( \gamma_n )$, there were additional
problems in these computations due to the accumulating errors from the
uncertainties in the values of zeros and $S(t)$.
Values computed over shorter intervals suggest that the mean values in Table~2.5.4 are accurate.
The entry for sign changes of $S_1 (t)$ refers to the number of sign changes
per Gram interval.
This figure appears to be moderately accurate.
Changing the initial value of $S_1 ( \gamma_n )$ by
$\pm 10^{-4}$ varied the number of computed sign changes of $S_1 (t)$ for the
$N=10^{20}$ interval only between 73799 and 74089.
\section{Extreme gaps between zeros}\label{sc25}
\hspace*{\parindent}
In its weakest form, the GUE hypothesis predicts only that \eqn{eq227} holds for all
$0 \le \alpha < \beta < \In$, and so it says nothing about the
existence of a small number $(o(M)$, say) of large or small $\delta_n$.
A double zero of the zeta function, giving $\delta_n =0$, would not by itself
contradict this weak hypothesis.
On the other hand, it is known (cf.~\eqn{eq223} and \eqn{eq225}) that in the GUE,
\beql{eq251}
Prob ( \delta_n \le x) = \frac{\pi^2}{9} x^3 - \frac{2 \pi^4}{225} x^5 +
\frac{\pi^6}{2205} x^7 + \cdots~ ,
\eeq
so very small $\delta_n$ (roughly $o(M^{-1}/3 )$ among $M$ samples) are
rare in the GUE, and a similar result holds for large $\delta_n$.
A strong form of the GUE hypothesis would predict that even extreme values of $\delta_n$ (and $\delta_n + \delta_{n+1}$) for the zeta function would behave
as in the GUE model.

Given the constraints on $S(t)$ described in Section~\ref{sc24}, we can expect that even if the strong
form of the GUE hypothesis holds, it only applies to the zeta function at large heights, and that the lower the region under investigation, the fewer extreme values of $\delta_n$ or
$\delta_n + \delta_{n+1}$ we will find.
This is clear for large values of $\delta_n$ and $\delta_n + \delta_{n+1}$, as these clearly
correspond to large values of $| S(t) | $.
It is also true for small values of $\delta_n$ and $\delta_n + \delta_{n+1}$, though, since several zeros clustered close together again force $|S(t)| $ to be large.

What was observed in \cite{Od2} in a comparison of the first $10^5$ zeros to $10^5$ zeros starting with zero number $10^{12}$ is that the above predictions
were largely satisfied by the data.
In general, there was a deficiency of extreme values of $\delta_n$ and
$\delta_n + \delta_{n+1}$
(compared to the GUE prediction),
but this deficiency declined as one considered the higher zeros.
There was, however, one observation that went counter to expectations.
The number of small $\delta_n$ that were observed at large heights
was larger than predicted by the GUE theory.
This excess was not large, but it was also observed in the data for
$10^5$ zeros starting with zero number $2 \times 10^{11}$, as well as by some data
based on the first $1.5 \times 10^9$ zeros.
This excess of small spacings was very counterintuitive, and so gave rise
to some suspicions about the validity of the GUE hypothesis.

Table~2.6.1 shows the extremal values of $\delta_n$ and $\delta_n + \delta_{n+1}$ that were found in each data set.
(The number of zeros in each data set is given in Table~1.2.)
The last column in Table~2.6.1 gives the probability that the minimal
$\delta_n$ would not exceed the values in the second column if all
the $\delta_n$ in the data set were drawn independently from the GUE
distribution.
From \eqn{eq251}, we see that the probability that the smallest $\delta_n$ out of $M$ that are drawn from the GUE satisfies $\delta_n \le x$ is
about
\beql{eq252}
1 - \lt 1 - \frac{\pi^2}{9} x^3 \rt^M \sim
1 - \exp ( - \pi^2 x^3 M/9 ) ~.
\eeq
This approximation was used to compute the last column of Table~2.6.1.
We can see that most of the entries in that column are high (although
not too high, which would indicate a severe deficiency of small spacings),
while those for $N=10^{18}$ (where $\delta_n = 0.001124$ for $n = 10^{18} + 12, 376, 780$, a case that will be discussed in sections~2.7 and 4.5) and for $N = 10^{19}$ (where $\delta_n = 0.000897$ for $n = 10^{19} + 15, 987, 196$ is the smallest $\delta_n$
that was found in our computations)
are extremely low.
The smallest value of $\delta_n$ that is known
is $\delta_n = 0.000310$ for $n = 1, 048, 449, 114$ (found by
van~de~Lune et~al. \cite{LRW2}),
and the probability of such a small spacing occurring among $1.5 \times 10^9$ samples drawn from the GUE is only 0.048.
Thus the extremely small values of the $\delta_n$ do appear to be somewhat
more frequent than expected.
(Some more evidence pointing to this conclusion is presented in Section~\ref{sc3}.)

When we next consider small, but slightly larger spacings, we find no evidence of an excess of small spacings.
Table~2.6.2 shows the number of $\delta_n \le 1/20$ and $\le 1/10$ observed in each set
(given in terms of cases per million zeros to make comparisons easier).
If we consider the $N=10^{19}$ entry for $\delta_n \le 1/20$, for example,
we see that we are dealing with 2353 cases altogether, so a normal
sampling error might be around 50, which is about 2\%.
Thus the 140.5 figure in the table is consistent with the 136.8 expected for the GUE.

Still another way to judge whether there is any anomaly in the distribution of the
$\delta_n$ or the $\delta_n + \delta_{n+1}$ is to
use
the quantile-quantile $(q-q)$ plots to compare
the observed distributions to those of the GUE.
Given a sample $x_1 , \dd , x_n$, and a continuous cumulative distribution
function $F(z)$ for some distribution, the $q-q$ plot is obtained by plotting
$x_{(j)}$ against $q_j$, where $x_{(1)} \le x_{(2)} \le \cdots \le x_{(n)}$ are the $x_i$ sorted in increasing order, and the $q_j$ are the theoretical
quantiles defined by $F(q_j ) = (j-1/2) /n$ \cite{CCKT}.
The $q-q$ plot is a sensitive method of
detecting differences among distributions.
In particular, while it does show the outliers that are far away from the expected position, it makes it possible to disregard them
and concentrate on the main part of the distribution curve.
If the $x_j$ are drawn from the distribution corresponding to $F(z)$, and the sample size $n$ is large, the $q-q$ plot will be close to the straight line $y=x$.
In all our $q-q$ plots, 
straight lines $y=x$ are drawn to facilitate comparisons.
(By the standards of typical statistical investigations, the sample sizes we deal with are very large,
and the degree of agreement between conjecture and 
numerical evidence is very good, so
one has to look at minute deviations.)

The $q-q$ plots of \cite{Od2} that showed the distribution of small $\delta_n$
indicated a deficiency of small $\delta_n$ for $N=1$, and a slight excess for
$N=2 \times 10^{11}$ and $N= 10^{12}$.
Those plots were each based on $10^5$ values of $\delta_n$.
When the new, more extensive data for $N=10^{12}$ were obtained, the resulting
$q-q$ plot was similar to that of Fig.~2.6.1, and did not behave like the plot in Fig.~8
of \cite{Od2} (which was based on only $10^5$ zeros).
Figures~2.6.1 and 2.6.2 show $q-q$ plots of $\delta_n$ drawn from two
disjoint sets of $10^6$ zeros near zero number $10^{20}$.
While the plot of Fig.~2.6.1 might indicate a slight excess of small
spacings (those in (0.02,0.04), roughly), and a slight deficiency of slightly larger spacings (where the scatterplot lies above the straight line),
Fig.~2.6.2 indicates almost perfect agreement between theory and experiment.
Figure~2.6.2 is not completely representative of zeros in the $N=10^{20}$ sets, since it was the one of several $q-q$ plots based on disjoint
sets of $10^6$ zeros that gave the best agreement.
Figure~2.6.1 is more typical in this respect.

Figures~2.6.1 and 2.6.2 provide only a little, if any,
support to the theory that there is an
excess of small spacings among the zeros.
Some further support can be found, however, if we combine all the data from the $N=10^8$, $10^{19}$, and $10^{20}$ data sets, which contain $112, 314, 006$ zeros, and yield 112,314,003 values of $\delta_n$.
The resulting $q-q$ plot, shown in Fig.~2.6.3, does indicate a slight excess of small
$\delta_n$ (the two outliers close to the bottom of the graph are the unusually small $\delta_n$ that
are minimal in the $N=10^{18}$ and $10^{19}$ data sets), but the evidence is not conclusive.

When we consider the other extremal values of $\delta_n$ and $\delta_n + \delta_{n+1}$, the evidence is in much better agreement with expectation.
The counts in Table~2.6.2 show that the numbers of small $\delta_n + \delta_{n+1}$, large $\delta_n$, and large $\delta_n + \delta_{n+1}$ are all smaller than predicted by the GUE theory, but increasing towards that prediction.
The $q-q$ plots of Figures~2.6.4 through 2.6.7 also support this impression;
there are too few extreme values in general, but the deficiency is smaller for $N=10^{20}$ than for $N=10^{12}$.

Because of \eqn{eq226}, one can expect that among the values of $\delta_n + \delta_{n+1}$ drawn from the GUE, the probability of the minimal value being $\le x$ is about
$$1- \exp ( - \pi^6 x^8 M/32400 ) ~.$$
The minimal value of $\delta_n + \delta_{n+1}$ of 0.1124 in the $N=10^{20}$ data set would then occur with a probability of 0.06 in the GUE, while the corresponding
probabilities for the $N=10^{12}$, $10^{14}$, $10^{16}$, $10^{18}$, and $10^{19}$ data sets are
0.93, 0.78, 0.25, 0.27, and 0.60, respectively.
Thus the only one of these figures that might seem unusually small is that
for the minimal $\delta + \delta_{n+1}$ for $N=10^{20}$.

The maximal values of $\delta_n$ and $\delta_n + \delta_{n+1}$ recorded
in Table~2.6.1 are all somewhat smaller than what the GUE predicts,
which is not too surprising given the bounds known to hold for $S(t)$ and
$S_1 (t)$.
For very large spacings in the GUE, des~Cloizeaux and Mehta \cite{CM2} have proved
that
\beql{eq253}
\log  p(0, t) \sim - \pi^2 t^2 /8 ~~~\mbox{as} ~~~ t \to \In ~,
\eeq
which suggests that
\beql{eq254}
\max_{N+1 \le n \le N+M}  \delta_n \sim \pi^{-1} ( 8 \log  M)^{1/2}
\eeq
as $N, M \to \In$ with $M$ reasonably large compared to $N$.
This is larger by about a $( \log  \log M)^{1/2}$ factor than the conjecture
\eqn{eq242}
of Montgomery
allows.
Our data are too limited to
decide
whether that conjecture is right.

Values of $\delta_n$ and $\delta_n + \delta_{n+1}$ larger than those of Table~2.6.1 have been found in other computations, and are described in Section~\ref{sc31}.
In particular, the largest known values of $\delta_n$ and of $\delta_n + \delta_{n+1}$ are 5.1454 and 6.0165, respectively.

Even on the assumption of the RH, it is only known that $\delta_n \le 0.5172$ and $\delta_n \ge 2.337$ each occurs infinitely often [CGG1], and $\delta_n \ge 2.68$ occurs infinitely often on the assumption of the Generalized Riemann Hypothesis for Dirichlet
$L$-functions (or at least of a Generalized
Lindel\"{o}f
Hypothesis) \cite{CGG2}.
On the assumption of the RH, it is also known that $\delta_n < 0.77$ and $\delta_n > 1.33$ each holds for a positive proportion of $n$ \cite{CGGGH}.
The GUE predicts that $\delta_n < \epsilon$ and $\delta_n > \epsilon^{-1}$ should each hold
for a positive proportion of $n$ for every fixed
$\epsilon > 0$.
If we could prove that $\delta_n < 1/4$ holds for
infinitely many $n$, we could obtain effective bounds for class numbers of
imaginary quadratic number fields \cite{MW}.
The GUE hypothesis predicts that $\delta_n < 1/4$ for 1.6\% of $n$'s, and this is very close to what is observed in numerical data.
(For $\delta_n < 1/2$ the corresponding figure is 11.3\%.)
\section{Long and short range correlations between zeros}\label{sc26}
\hspace*{\parindent}
The distribution of the eigenvalue
spacings in the GUE
is stationary.
What this means is that for any $k$, the frequency with which $( \delta_n$,
$\delta_{n+1} , \dd , \delta_{n+k} ) \in Q$ for any measurable subset $Q \subseteq \RR^{k+1}$ does not depend on the range of $n$, so that the distribution eigenvalue spacings looks every place the same.
On the other hand, this distribution is not Markovian, so that the distribution
of $\delta_{n+1}$ does not depend just on that of $\delta_n$.
Instead, $\delta_{n+1}$ is correlated to all the neighboring $\delta_n$,
$\delta_{n-1} , \dd$, as well as $\delta_{n+2}$, $\delta_{n+3} , \dd$.
In the limit, that also should be true for the zeros of the zeta function.
However, given the slow growth rate of $S(t)$, one cannot expect GUE
behavior from joint distributions of
$\delta_n , \delta_{n+1} , \dd , \delta_{n+k}$ if $k$ is
large.
Already the data of sections~2.3 and 2.5 show that the behavior of $\delta_n$ is much closer to the GUE prediction than that of $\delta_n + \delta_{n+1}$.
That was the main reason for not investigating $\delta_n + \delta_{n+1} + \delta_{n+2}$ and higher order spacings.

When we investigate long range correlations among the zeros of the zeta function,
we find phenomena connected not to the GUE, but to the
distribution of primes.
For example, if we let the autocovariances of a set of $\delta_n$ be defined by
\beql{eq261}
c_k = c_k (H, M) = \frac{1}{M}  \sum_{n=H+1}^{H+M}
( \delta_n -1) ( \delta_{n+k} -1 ) ~,
\eeq
then it has been conjectured by F.~J. Dyson (unpublished) that in the GUE,
\beql{eq262}
c_k \approx  - \frac{1}{2 \pi^2 k^2}
\eeq
for $k > 0$, with the $approx$ indicating some degree of approximation,
not necessarily asymptotic equality as $N, M \to \In$.
This result has not been proved for the GUE, but it is intuitively appealing for both the GUE
and the zeros of the zeta function, since it says in effect that a large spacing would lead to
smaller spacings nearby (and vice versa), and that this effect would diminish as one
considered spacings further and further apart.

What was observed in \cite{Od2} for the $\delta_n$ was quite different from the conjecture \eqn{eq262}.
Additional data based on the new computations are presented in
Table~2.7.1.
The $N=1$ entries come from the \cite{Od2} computations, and have $H=0$, $M=10^5$.
The $N=10^{12}$ and $10^{20}$ entries come from the new computations,
and both have $M=10^6$,
with $H = 10^{12} - 6, 032$ for the $N=10^{12}$ column and $H=10^{20} - 48, 776$ for the $N=10^{20}$ column.
(A comparison of the $N=10^{12}$ entries here with those in Table~6 of \cite{Od2}, which are based on 1/10 as many zeros indicates
the size of the sampling errors.)
For small $k$, the data in this table support Dyson's conjecture
\eqn{eq262}.
For higher sets of zeros,
the agreement with \eqn{eq262} extends to slightly higher values of $k$.
However, for large $k$, we see totally different behavior.
If $\delta_n$ and $\delta_{n+k}$ were independent, then, since their mean value
is 1 and variance is about 1/6, we would expect a sum of $10^6$ terms of the form
$( \delta_n -1) ( \delta_{n+k} -1 )$ (for $k > 0$) to be about
$10^{6/2} / 6 \approx 170$, and this would correspond to a value of $c_k$ of
$1.7 \times 10^{-4}$.
The values in Table~2.7.1 for $9,980 \le k \le 10,000$ are usually much larger than that, which shows that there are strong long range
correlation between the $\delta_n$.
The pattern of signs of the $c_k$ also shows the nonrandom
nature of the $c_k$.
The $c_k$ are occasionally positive, and occasionally negative, indicating that for some $k$, a large $\delta_n$ tends to be associated with large $\delta_{n+k}$, while for other $k$ it tends to be associated with small $\delta_{n+k}$.

An explanation for the long range dependencies among the $\delta_n$ was proposed in \cite{Od2}.
It implies that the observed correlations come from primes through
formulas such as that of Landau \cite{Lan1}, which says that for any fixed $y > 0$, as
$N \to \In$ we have
\beql{eq263}
\sum_{n=1}^N  e^{i \gamma_n y} =
\left \{
\begin{array}{ll}
- \df{\gamma_N}{2 \pi} e^{-y/2} \log p + O(e^{y/2} \log N) &
\mbox{if}~~y = \log p^m ~, \\
~~ \\ [-.08in]
O(e^{-y/2} \log N) &
\mbox{if}~~ y \neq \log p^m~,
\end{array}
\right.
\eeq
where $p$ denotes a prime and $m \in Z^+$.
The above statement assumes the RH, but Landau proved a similar unconditional result.
Improvements on Landau's result (with better error terms and more explicit dependencies of the error terms on $y$) have been obtained
by Fujii \cite{Fu5,Fu7} and
Gonek \cite{Gon2}.
(There are many formulas relating primes and zeros, and the ``explicit formulas'' of Guinand \cite{Gu1,Gu2} and Weil \cite{We1} are among the most general.)

The paper \cite{Od2} presents the detailed explanation of how Landau's formula \eqn{eq263} forces the spectrum of the $\delta_n$ to consist largely of point masses at frequencies corresponding to prime powers, which then
forces the initially unexpected behavior of the $c_k$ that is seen in the tables.
This explanation will not be repeated here.
We will mention only that while it is not rigorous, it is supported
by heuristics and numerical evidence.
What we will do now is to check how well Landau's formula \eqn{eq263}
fits with the numerical data.
The main interest here is to see just how many zeros $\gamma_n$ are needed at
various heights to observe the phenomenon of large values
occurring at logarithms of prime powers.
Some proposals have even been made to use sums
like that in \eqn{eq263} for
primality testing and factoring integers.
While it seems unlikely that efficient methods could be developed by this approach,
it is of some interest to see what happens when one considers a relatively
short sum over high zeros.

Let
\beql{eq264}
h(y) = \sum_{n=10^{20} +1}^{10^{20} +4 \times 10^4}  e^{i \gamma_n y} ~.
\eeq
Figure~2.7.1 shows a graph of $2 \log  | h(y) | $ for $0 \le y \le 3$.
It is instructive to compare this graph with that of Fig.~15 of \cite{Od2},
which is drawn on the same scale, but is based on an exponential sum of
$4 \times 10^4$ zeros starting at zero number $10^{12} +1$.
Both graphs show sharp peaks precisely at logarithms of prime powers,
and the peaks are visibly higher at primes than at proper prime powers, as predicted by Landau's formula.
(The heights of the peaks are not represented too accurately on the graph because of
limited sampling.)
All the prime powers $< e^3 = 20.09$ are visible.
The main difference between the two graphs is that in Fig.~2.7.1 the peaks are
slightly lower, and the ``noise'' region between the peaks has somewhat
higher values.
Furthermore, the regular patterns seen in the ``main'' regions of
Fig.~15 of \cite{Od2} (which come from sampling at regular intervals
a rapidly oscillating function whose frequency and amplitude are
changing slowly) is not visible in Fig.~2.7.1.
These differences are probably due partly to the errors in the computed
values of the $\gamma_n$ near the $10^{20}$-th zero and partly
to
taking a very short sum.
$4 \times 10^4$ zeros out of the first $10^{20}$ is a very small
proportion, so it is remarkable that the pattern of Fig.~2.7.1 is as clear
as it is, since this is much better than the proved results of
\cite{Fu5,Fu7,Gon2,Lan1} might lead one to expect.

Figure~2.7.2 shows a graph of $2 \log | h(y) | $, where $h(y)$ is again defined
by \eqn{eq264}, but this time over the region $8 \le y \le 8.05$.
(This graph is based on $10^4$ equally spaced values of $y$.)
The interval from $e^8 = 2980.96$ to $e^{8.05} = 3133.79$ contains the primes
2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, and 3121,
and the prime power $5^5 =3125$.
Figure~2.7.2 fails to distinguish between several close pairs
of primes.
When one graphs a similar sum, but with 10 times as many zeros,
as is done in Fig.~2.7.3, all the primes can be distinguished,
and even 3125 can be easily discerned.

An elegant measure of
long range correlations between zeros was found by
Berry \cite{Be4}.
If we consider an interval of length $2 \pi L ( \log  (T/(2 \pi ))^{-1}$ at height $T$,
the expected number of zeros in it equals $L$.
We define the number variance of the zeros by
\beql{eq265}
V_T (L) = V_{T,H} (L) =
H^{-1} \int_T^{T+H} \left\{ N \lt t + \frac{2 \pi L}{\log (t/(2 \pi ))} \rt - N(t) - L \right\}^2 dt ~.
\eeq
In the GUE, one has $V_T (L) = G(L)$, with
\begin{eqnarray}
\label{eq266}
\lefteqn{G(L) = \pi^{-2} \{ \log (2 \pi L) - Ci( 2 \pi L ) - 2 \pi L Si (2 \pi L )} \nonumber \\
&&~~~~~~~~~~~~~~~~+~\pi^2 L - \cos ( 2 \pi L ) + 1 + c_0 \} ~,
\end{eqnarray}
where $Ci$ and $Si$ are the cosine and sine integrals \cite{HMF} and $c_0 = 0.577 \dd$
is Euler's constant.
Asymptotically,
\beql{eq267}
G(L) \sim \pi^{-2} \log (2 \pi L ) ~~~~\mbox{as} ~~~~ L \to \In ~,
\eeq
while
\beql{eq268}
G(L) \sim L ~~~~\mbox{as} ~~~~ L -> 0 ~.
\eeq
Gallagher and Mueller \cite{GM} showed that Montgomery's pair correlation conjecture
implies $V_T (L) = L-L^2 + o( L^2 )$ as $L \to 0$, which is
consistent with \eqn{eq268}.
(See also \cite{Fu8}.)
On the other hand,
the numerical evidence of \cite{Od2} showed that $V_T (L)$ was small even for
moderately large $L$, and so a relation like \eqn{eq267} appeared impossible.
Motivated by this discovery,
by the relations between primes and long range correlation between zeros
discussed above, and by his earlier work on eigenvalues of Hamiltonians of
chaotic dynamical systems \cite{Be1,Be2,Be3}, Berry \cite{Be4} found heuristic arguments
which suggested that for any $\tau \in (0, 1)$, and any $L > 0$,
\beql{eq269}
V_T (L) \approx G(L) + B_T (L) ~,
\eeq
where for $U = T (2 \pi )^{-1}$,
\beql{eq2610}
B_T(L) = \pi^{-2}
\left\{ 2
\begin{array}{c}
_{~~} \\ \dis\sum_{p} \dis\sum_{r=1}^\In \\ ^{p^r < U^\tau}
\end{array}
\df{\sin^2 ( \pi Lr( \log p ) / ( \log U))}{r^2 p^r}
+ Ci(2 \pi L \tau ) - \log (2 \pi L \tau ) - c_0 \right\}\,,~~~~~~~~~~
\eeq
and $p$ denotes primes.
Computations using $10^5$ zeros near zero number $10^{12}$, using values of $L$
up to 1000, showed excellent agreement between Berry's conjecture \eqn{eq269} and empirical data, and those results are shown in the graphs in \cite{Be4}.
Note that the $\log (L)$ terms in $G(L)$ and $B_T (L)$ cancel out, and so for every fixed $L$,
one can show that there is a positive function $g(L)$ such that
\beql{eq2611}
G(L) + B_T (L) = g(L) + o(1) ~~~~\mbox{as} ~~~~ T \to \In ~.
\eeq
Moreover, if $\tau$ is held fixed, then it is easy to see that
\beql{eq2612}
G(L) + B_T (L) \sim \pi^{-2} \log \log T ~~~~\mbox{as} ~~~~
T~, L \to  \In ,
\eeq
(with $L$ growing much more slowly than $T$),
since the arguments of the sine in the definition \eqn{eq2610} of $B_T (L)$ will
be asymptotically equidistributed modulo $2 \pi$.

The new zeros were used to obtain further data.
For $N=10^{12}$, the number variance $V_T (L) = V_{T,H} (L)$ defined
by \eqn{eq265} was computed with
$$
\begin{array}{r@{~}l@{~}l@{~~~~~~~~~~~~~~~~~~}r@{~}l@{~}l}
T & = & \gamma_{n_0} , & n_0 & = & 10^{12} - 6, 032~, \\
~~ \\ [-.09in]
T+H & = & \gamma_{m_0} , & m_0 & = & n_0 + 5 \times 10^5 ~.
\end{array}
$$
For $N=10^{20}$, the values that were chosen were
$$
\begin{array}{r@{~}l@{~}l@{~~~~~~~~~~~~~~~~~~}r@{~}l@{~}l}
T & = & \gamma_{n_1}, & n_1 & = & 10^{20} - 48 , 778 ~, \\
~~ \\ [-.09in]
T+H & = & \gamma_{m_1} , & m_1 & = & n_1 + 5 \times 10^5 ~.
\end{array}
$$
Berry's function \eqn{eq2610} was computed in each case with $\tau = 1/4$.
(Varying $\tau$ between 0.2 and 0.3 did not appreciably change the results,
as was to be expected.)
The results of some of these computations for $N=10^{20}$ are presented
in Figs.~2.7.4 through 2.7.6.
In Fig.~2.7.4, the dashed line is the graph of the GUE prediction $G(L)$,
the solid line is the graph of Berry's prediction $G(L) + B_T (L)$, and
the scatterplot is that of computed values of $V_T (L)$.
In Figs.~2.7.5 and 2.7.6 the graphs of the computed values of
$V_T (L)$ and of Berry's prediction $G(L) + B_T (L)$ were both drawn as solid lines, one superposed on the other.
The slight differences between the two curves show up as slight blotches on the graph.
(The empirical data is slightly more wiggly than $G(L) + B_T (L)$.)
We see that even for $L = 5 \times 10^5$, the agreement between
computed and predicted values is almost perfect.

A comparison of the graphs of \cite{Be4} (and of similar graphs drawn with the more extensive data that has been obtained
for $N=10^{12}$ in the present computation) with Figs.~2.7.4 through 2.7.6 shows that for $N=10^{20}$, the number variance oscillates less than for $N=10^{12}$.
The agreement of data with Berry's prediction is better for $N=10^{20}$.

While Berry's prediction \eqn{eq269} for $ V_T (L) $was based
on heuristic arguments, one can prove that
a version of the conjecture follows from
the RH and the pair correlation
conjecture \eqn{eq2211}.
This will be shown in a
separate paper \cite{Od4}.
\section{Lehmer phenomenon}\label{sc27}
\hspace*{\parindent}
For the RH to be true,
$|Z(t)| $ cannot have any relative minima between two
consecutive zeros
of $Z(t)$.
Cases where
\beql{eq271}
v_n = \max_{\gamma_n < t < \gamma_{n+1}}  |Z(t)|
\eeq
is very small (so that in a sense the RH is ``almost violated'') are referred to as
Lehmer's phenomenon \cite{Lr2}, and provide some of the more interesting heuristics both for and
against the RH (cf.~\cite{Od3}).
In this section we present statistics on the frequency of this phenomenon (which
does not have a precise definition).

The zero-locating program printed the largest value of $|Z(t)| $ that had been computed in each stretch of $10^4$ zeros.
To provide further information, the program was modified for the $N=10^{19}$, $N=10^{20}$, and $N = 2 \times 10^{20}$ data sets to obtain
statistical information about the behavior of
$v_n$.
Since getting a very good approximation to $v_n$ would have required
substantial computing time, what the program computed was the midpoint value
\beql{eq272}
w_n = |Z(( \gamma_n + \gamma_{n+1} ) / 2) | ~.
\eeq
When a value of $w_n > 250$ or $w_n < 5 \times 10^-4$ was encountered, it was printed together with $n$,
$\gamma_n$, and $\delta_n$.
(However, $w_n$ was not computed for a total of roughly 100 zeros at the ends of data sets.)

To see how good an approximation $w_n$ was to $v_n$, the values
\beql{eq273}
v_n^{\ast} = \max_{1 \le k \le 39} \left | Z \lt \gamma_n + \frac{k}{40}
( \gamma_{n+1} - \gamma_n ) \rt \right |
\eeq
as well as of $w_n$ were computed for
$n_0 \le n \le n_0 + 8 \times 10^5 -1$, where $n_0 = 10^{20} + 15, 316, 087$.
Let
\beql{eq274}
r_n = v_n^{\ast} / w_n ~.
\eeq
Then the maximal value of $r_n$ that was found was 1.43.
Only 755 out of the $8 \times 10^5$ values of $r_n$ were $> 1.2$, while the rms
value of $r_n -1$ was 0.029.
Among the 873 values of $n$ for which $\delta_n < 0.1$, the maximal
value of $r_n$ was 1.008, and the rms value of $r_n -1$ was
$5.1 \times 10^-4$.
For the 898 values of $n$ for which $\delta_n > 2.5$, the corresponding
numbers were 1.29 and 0.036.
For the 244 values of $n$ for which $w_n > 100$, these numbers
were 1.137 and 0.032, while for the 1426 values of $n$ for which
$w_n < 0.01$, they were 1.072 and 0.0066.
Thus in general the values of $w_n$ do provide good approximations to $v_n^{\ast}$, and therefore surely also to $v_n$.
This was to be expected on the basis of the GUE predictions (in particular that
the approximation would be exceptionally good when $\delta_n$ is small).
The size of $v_n$ is determined largely by the few zeros
nearest to $\delta_n$ (cf.~\cite{Hej5,Hej6}), and so under the assumption
of the GUE one can make quantitative predictions about the behavior of $r_n$.

Table~2.8.1 shows the frequency of occurrence of values of $w_n < 5 \times 10^-4$ among the approximately
$3 \times 10^8$
values of $n$ that were checked in the
$N=10^{19}$, $N=10^{20}$, and $N= 2 \times 10^{20}$ data sets.
The smallest value of $w_n$ that was found there was $1.82 \times 10^{-6}$,
for $n=10^{20} + 52, 127, 155$ and
$\delta_n = 0.00263$,
while the second smallest was $1.84 \times 10^{-6}$.

One might expect, and one does observe empirically, that the Lehmer phenomenon
is associated to small values of $\delta_n$.
If $\delta_n$ is small, then one might expect that $w_n$ is almost
proportional to $\delta_n^2$, since zeros other than $\gamma_n$ and
$\gamma_{n+1}$ ought to contribute multiplicative factors that
behave like a power of $\log \gamma_n$ on the average, and are at most
$\gamma_n^{o(1)}$ as $n \to \In$ (assuming the
Lindel\"{o}f conjecture).
Since the probability that $\delta_n \le x$ is about $\pi^2 x^3 /9$ for $x$ small (see Sections~\ref{sc22} and \ref{sc25}), one might conjecture that the probability
of $w_n < y$ might be proportional to $y^{3/2}$.
This would suggest that among the first $n$ zeros, the smallest $w_n$
might be
$n^{-2/3}+o(1)$ as $n \to \In$.
If true, this relation would settle an old question \cite{Ed} about the number of terms in the asymptotic part of the Riemann-Siegel formula that have to be used to
separate the zeros;
even the old estimate of Titchmarsh \cite{Tit1} with an error term of $O(t^{-3/4} )$ would suffice at large heights.

The above heuristic about the behavior of small $w_n$ is supported well by
empirical data.
Let $W$ denote the set formed by the $N= 10^{19}$ data set and the first 78,893,234 zeros in set $N= 10^{20}$.
(In the notation of Table~4.7.1, it is the union of sets $i$, $k$,
$l$, $m$, and $n$.)
In set $W$, 1976 values of $n$ have
$w_n < 5 \times 10^-4$.
Among these 1976 values,
the ratio $w_n / \delta_n^2$ varies between 0.0136 and 8.56, with a mean of 0.608 and a variance of 0.427.
Thus the correlation between $\delta_n^2$ and $w_n$ is only modest.
On the other hand, these $w_n$ follow almost perfectly the rule
conjectured above that the fraction of them that are $< y$ ought to be proportional
to $y^{3/2}$.
This can be seen
by looking at the ratio of the
$k$-th smallest $w_n$ to $5 \times 10^{-4} \times (k/1976)^{2/3}$, which varies between 0.715 and 1.267, with a mean of 1.01 and a variance of $9 \times 10^{-4}$, and from looking at a $q-q$ plot of the sorted $w_n$ against
$5 \times 10^{-4} \times (k/1976)^{2/3}$.
We also find good agreement between this prediction of the behavior of the small $w_n$ and the counts of Table~2.8.1, which are based on all of the zeros in the $N= 10^{19}$,
$10^{20}$, and $2 \times 10^{20}$ data sets.
Thus on average the influence of the neighboring $\gamma_k$ cancels out.

The most extreme example of the Lehmer phenomenon that was found during
the computations described in this paper occurs for $n= 10^{18} + 12, 376, 780$, where
$w_n = 5.28 \times 10^{-7}$ and $\delta_n = 0.001124$.
A graph of $Z(t)$
near
this point is given in Fig.~2.8.1.
(Figure~2.8.1 shows also what looks like another case of the Lehmer phenomenon near Gram point $n-5$, but in that case the minimum of $Z(t)$ reaches $-0.0094$, and so it does not
qualify under our definition.)
A much more detailed view of $Z(t)$ in a small neighborhood of this Lehmer
phenomenon is given in Fig.~4.7.1.
(That picture plays an important role in the discussion of the validity of the present
computations that is presented in Section~\ref{sc45}.)

The most extreme example of the
Lehmer
phenomenon
that is known was found by van~de~Lune et~al. \cite{LRW2}.
For $n=1, 048,449, 114$, they discovered that $\delta_n = 0.000310$, while
$v_n = 2.2 \times 10^{-7}$ $(\ge w_n )$.
Since the height of this example is only about the square root of that
for $n=10^{18} + 12, 376, 780$, it could be
argued that the higher example of this paper is even more extreme.
However, the $\delta_n$ found by van~de~Lune et~al. is by far the smallest of any that are known.
%\section{Large values of $\mbox{\protect\boldmath $\zeta$}$ {\bf (1/2+it)}}
\section{Large values of $\zeta (1/2+it)$}\label{sc28}
\hspace*{\parindent}
The largest value of $|Z(t) | = | \zeta (1/2 + it )| $ that was encountered by van~de~Lune et~al. \cite{LRW2} in their investigation of the first $1.5 \times 10^9$
zeros was 117.
Table~2.9.1 lists the largest values of $|Z(t)| $ that were encountered in each of the
data sets computed in this paper.
The main zero locating program kept track of the largest value of $|Z(t)| $ that had been
computed, but did not attempt to do a systematic search for large values.
However, since large values are usually
associated with large $\delta_n$, the standard zero locating procedure seemed to be
quite good at finding the high peaks in $|Z(t)| $.
For the $N=10^{19}$, $N=10^{20}$, and $N= 2 \times 10^{20}$ data sets, the more careful procedure
described in Section~\ref{sc27} was employed,
which provided even more reliable statistics.
The number of values of $n$ in those two data sets for which
$w_n$ (defined by Eq.~\eqn{eq272}) exceeded various
thresholds is given in Table~2.9.2.
(Section~\ref{sc3} lists some
values of $t$ for which $|Z(t)| $ is much larger
and which were found by a different procedure.)

The rate of growth of $|Z(t)| $ is one of the most intensively studied problems in the theory
of the zeta function, since bounds on it provide estimates on the
density of possible
zeros away from the critical line.
It is easy to show that
\beql{eq281}
|Z(t)| \le t^{\alpha + o(1)} ~~~\mbox{as} ~~~ t \to \In
\eeq
with $\alpha = 1/4$.
Exponential sum methods were used in the first few decades of this
century to show that \eqn{eq281} holds with $\alpha = 1/6$,
and then to successively lower this value of $\alpha$.
(See the Notes to Chapter~5 of \cite{Tit2} for a list of
the improvements.)
Until recently, the smallest value of $\alpha$ for which \eqn{eq281} was known to hold was $\alpha = 139/858 = 0.162004 \dd$,
proved by Kolesnik \cite{Ko}, and there were indications that this result
was close to the limit of what the two-dimensional ``exponent pair'' method that was being used could yield \cite{GK}.
However, Bombieri and
Iwaniec \cite{BI} have obtained a new method that gave $\alpha = 9/56 = 0.16071 \dd$.
This method was then developed by Huxley,
Kolesnik, and Watt in a series of papers,
and the latest result, proved by Huxley \cite{Hux} is that \eqn{eq281}
holds with
$\alpha = 89/570 = 0.15614 \dd$.

The Lindel\"{o}f hypothesis is the
statement that \eqn{eq281} holds with $\alpha =0$.
The RH yields a slightly stronger bound \cite{Tit2}
\beql{eq282}
|Z(t)| ~\le~ \exp (c( \log  t) ( \log  \log  t)^{-1} )
\eeq
for some $c>0$.
On the other side,
Balasubramanian and Ramachandra \cite{Bala,BR} have
shown that
\beql{eq283}
\max_{0 \le t \le T}  |Z(t) | \ge \exp \lt
\frac{3( \log  T)^{1/2}}{4( \log  \log  T)^{1/2}} \rt
\eeq
if $T$ is large enough and more generally, that if $\eta > 0$,
then for $T \ge T( \eta )$ and
$( \log  T)^\eta \le H \le T$, we have
\beql{eq284}
\max_{T \le t \le T+H} |Z(t)| \ge
\exp \lt
\frac{3( \log  H)^{1/2}}{4( \log  \log H)^{1/2}} \rt ~.
\eeq
Montgomery \cite{Mon3} has conjectured that \eqn{eq283} is close to the real
rate of growth of $|Z(t)| $.

While the data that was collected about large values of $|Z(t)| $ probably
does reflect accurately the behavior of the zeta function in these ranges,
it does not help in assessing what the true rate of growth of $|Z(t)| $ is.
There are two main problems.
One is the relatively small number of zeros that were investigated.
Since large values of $|Z(t)| $ are rare, we probably do not even have a
good representation of the large values of $|Z(t)| $ for
$t < \gamma_n$, $n=10^{20}$.
(This is supported by the results of Section~\ref{sc3}, where much higher values were
found by special methods.)
Another problem in using our data to assess the true growth rate of $|Z(t)| $
arises from the slow approach to its true asymptotic behavior.
As is noted in Section~\ref{sc211} (see especially Fig.~2.10.1), even $\log |Z(t)| $ in the ranges
that have been investigated can be far from its eventual distribution.
Furthermore, as was noted in the Introduction, even when one investigates at heights
$t \approx 1.5 \times 10^{19}$, it is hard to tell the differences
in growth rates between
various functions.
(The situation is not as bleak as might seem from the argument used in the Introduction,
since one can use sensitive tools such as ratios of values of a function
at different points to estimate its growth rate, but that only helps to a limited extent.)
Note that for $t= \gamma_n$, $n=10^{20}$,
the bound
\eqn{eq283} is only 12.9.

Before concluding this section, we present some more statistics on the large values of
$|Z(t)| $ that were found in the
$W$ data set, which was defined in Section~\ref{sc27}, and which is a subset of the
$N= 10^{19}$ and $10^{20}$ data sets.
In set $W$,
565 values of $n$ were recorded for which $w_n > 250$.
The largest is $w_n = 631.7$ for $n = 10^{20} + 13, 704, 916$, for which
$\delta_n = 3.1428$.
(The maximum of $|Z(t)| $ between $\gamma_n$ and $\gamma_{n+1}$ is at least 641,
and there is no violation of Rosser's rule
near
$\gamma_n$.)
Of the 565 values, 94 are associated with violations of Rosser's rule.
(Of the 28 values of $n$ for which $w_n > 400$, 7 are associated with violations of
Rosser's rule.)
The smallest value of $\delta_n$ that was found for these 565 values of $n$ was
2.07, and the largest was 4.03.

There was a substantial correlation between $\delta_n^2$ and $w_n$ among these 565 samples in set $W$.
The ratio $w_n / \delta_n^2$ was in the range
(19.47, 64.74), with a mean of 35.47 and variance 61.32.
However, at large heights one would expect this correlation to diminish, in contrast to the situation
for Lehmer's phenomenon (Section~\ref{sc27}).
In the latter case the GUE theories predict that $\delta_n^2$ will
occasionally get as small as $n^{-2}/3$, so that the influence of the other zeros (likely to be $n^{o(1)}$ because of the
Lindel\"{o}f hypothesis and the separation of the other zeros
that is
predicted by the GUE) will not affect the size of $Z(t)$ very much.
On the other hand, the GUE theories predict that $\delta_n^2 = O ( \log  n )$, and since $Z(t)$ is known to get much larger (cf.~\eqn{eq283}), this must be due to some
long-range imbalances in the locations of the zeros.
One model for the distribution of $Z(t)$ (first proposed informally by Montgomery,
and worked out in detail by
Bombieri and Hejhal
\cite{BH,Hej5,Hej6}) predicts that at large heights, the size of $Z(t)$ is determined primarily by long ``amplitude'' waves, which are then modulated by local
distributions of zeros.
This model predicts that there should be clusters of large values of $|Z(t)| $, and
that over wide ranges, $w_n$ ought to depend mostly on the ``amplitude'' waves, and not on $ \delta_n$.
That there is a very strong correlation between the large $w_n$ and $\delta_n^2$ in our data might therefore indicate that we are not yet seeing the true asymptotic behavior.
%\section{Moments of $\mbox{\protect\boldmath $\zeta$}$ {\bf (1/2 + it)}}
\section{Moments of $\zeta (1/2 + it)$}\label{sc29}
\hspace*{\parindent}
It is conjectured that for every $\lambda \ge 0$,
\beql{eq291}
\lim_{T \to \In}  T^{-1} ( \log T)^{- \lambda^2}
\int_0^T  | Z(t) |^{2 \lambda} dt = c( \lambda )
\eeq
exists, with $c( \lambda ) > 0$ for all $\lambda$.
A proof of this conjecture, or even of some much weaker bound, would be very important, since it would prove the Lindel\"{o}f
conjecture.
However, this conjecture is only known to be true for $\lambda =0$ with
$c(0)=1$ (trivial), $\lambda =1$ with $c(1)=1$, and $\lambda =2$ with $c(2) = (2 \pi^2 )^{-1}$ (see the Notes for Chapter~7 in \cite{Tit2} for detailed information and references).
No specific values have been conjectured for $c( \lambda )$ in general,
but under the assumption of the RH, Conrey and Ghosh \cite{CG1} have shown that
$c( \lambda ) \ge c_1 ( \lambda )$, where
\beql{eq292}
c_1 ( \lambda ) = \Gamma ( 1+ \lambda^2 )^{-1} 
\prod_p \left \{ \lt 1 - \frac{1}{p} \rt^{\lambda^2}
\sum_{m=0}^\In ~ \lt \frac{\Gamma (m+ \lambda )}{m! \Gamma ( \lambda )} \rt^2 p^-m \right \} ~,
\eeq
and since $c( \lambda ) = c_1 ( \lambda )$ for $\lambda =0$ and 1, they suggested that perhaps $c( \lambda ) = c_1 ( \lambda )$ for all $\lambda \in [0, 1]$.
Since $c_1 (2) = ( 4 \pi^2 )^{-1} = c(2)/2$, equality of $c( \lambda )$ and
$c_1 ( \lambda )$ is unlikely outside the range
$0 \le \lambda \le 1$.
(There is a mistake on this point in the Notes to Chapter~7 of \cite{Tit2}.)
Conrey and Ghosh \cite{CG3} have shown that the derivatives of $c_1 ( \lambda )$ and $c( \lambda )$ with respect to $\lambda$ agree at $\lambda =0$ and 1.
Also, for $0 \le \lambda < 2$, Heath-Brown [HB2] has shown under the assumption of the RH that if $c ( \lambda )$ exists, it is not much larger than predicted by the Conrey-Ghosh conjectures.

One purpose of this section is to provide some numerical evidence about possible values of $c( \lambda )$.
One might expect that if
\beql{eq293}
r( \lambda , T, H) = H^{-1} ( \log  T)^{- \lambda^2}
\int_T^{T+H}  |Z(t)|^{2 \lambda} dt ~,
\eeq
then $r( \lambda , T, H) \sim c( \lambda )$ as $T \to \In$,
if $H$ grows sufficiently fast with $T$ while $\lambda$ is held fixed.
Table~2.10.1 presents some values of $r( \lambda , T, H)$
computed for $T= \gamma_{n_0}$ with $n_0 = 10^{20} + 47, 098, 588$ and
$T+H = \gamma_{n_1} , n_1 = n_0 + 10^6$.
Each of the $10^6$ gaps between consecutive zeros was divided into 40
intervals, $Z(t)$ was evaluated at the endpoints of these subintervals,
and Simpson's rule was applied to estimate the integral.
Variations on this procedure showed that it produced estimates that were
accurate to at least three decimal places (and more for high moments, as
Simpson's rule is least accurate for small $\lambda$, where the function has
singularities at zeros
that are hard to deal with).
However, the values in the tables, especially for large $\lambda$, have to be used with
caution because even an interval of $10^6$ zeros around zero number $10^{20}$ is too small
to be truly representative.
For example,
similar data was obtained for
$T= \gamma_{n_2}$ with $n_2 = 10^{20} + 15, 316, 087$ and
$T+H = \gamma_{n_3}$, $n_3 = n_2 + 8 \times 10^5$, and also for
$T= \gamma_{n_4}$, $n_4 = 10^{20} - 15, 409, 244$,
$T+H = \gamma_{n_5}$, $n_5 = n_4 + 10^6$.
For $\lambda =1$, the values found there differed by less than
0.5\% from those in Table~2.10.1, but for $\lambda = 2.5$ these values were 1.20
and 0.752 times those in Table~2.10.1, respectively.
The problem is that high
moments are determined largely by the few exceptionally large values of $Z(t)$, and those are
rare.
(See the next section for some further evidence of this.)
To get a good sample, for large $\lambda$, one would
need to integrate $|Z(t)|^{2 \lambda}$ over much longer intervals.

The data in Table~2.10.1 are consistent with the Conrey-Ghosh
conjectures that $c( \lambda ) = c_1 ( \lambda )$ for $0 \le \lambda \le 1$.
However, given the differences between the empirical data for $\lambda =1$ and $\lambda =2$ and the known asymptotic values, it is hard to draw any
definitive conclusions.
For $\lambda =1$, estimates of the second moment of $Z(t)$ are known
that are better than \eqn{eq291}.
They are of the form
\beql{eq294}
\int_0^T ~Z(t)^2 dt = T( \log  T -1- \log (2 \pi ) + 2 c_0 ) + E(T) ~,
\eeq
where $c_0$ denotes Euler's constant
$(= 0.577215 \dd)$, and $|E(T)| = O(T^\alpha )$ for various $\alpha < 1/3$.
(The best current value of $\alpha$ is $139/429 + o(1)$ as $T \to \In$,
due to Kolesnik \cite{Ko} and in a slightly sharper form to Hafner and
Ivi\'{c} \cite{HI}.
Note that $139/492 = 0.3240 \dd$)
If we let $r^{\ast} ( \lambda , T, H)$ be defined similarly
to $r( \lambda , T, H)$, but with $\log T$ in \eqn{eq293} replaced by $\log T - \log ( 2 \pi ) + 2 c_0$, we find that for the values of $T$ and $H$ that were used to
compute Table~2.10.1, $r^{\ast} (1, T, H) = 1.004$, which is closer to the asymptotic value $c(1) =1$ than the value of
$r(1, T, H) = 0.989$.
(The other two sets of values that were considered give $r^{\ast} (1, T, H) = 1.0003$ and 0.9995, respectively.)
Thus a major
problem in using empirical data is that we do not have good conjectures about asymptotics of moments of $Z(t)$, and that second order terms in those asymptotics are likely to be only slightly smaller
than the main terms.
(See also Section~\ref{sc210} on deviations between observed and expected behavior of $Z(t)$.)

Some data were obtained also about the negative moments of $|Z(t)| $.
Table~2.10.2 shows some values of
$$\frac{1}{H} \int_T^{T+H}
|Z(t)|^{- 2 \lambda} dt$$
for $T$ and $H$ as in Table~2.10.1.
(The values for $T= \gamma_{n_2}$, $T+H = \gamma_{n_3}$,
were essentially identical.)
They were obtained by applying Simpson's rule to the
inner 38 subintervals in every gap between consecutive zeros, and approximating $|Z(t)| $ by a linear function on the two outer subintervals.

Conrey and Ghosh \cite{CG2} have shown (assuming the RH) that
\beql{eq295}
\frac{1}{M}
\sum_{m=1}^M \max_{\gamma_m < t < \gamma_m+1} Z(t)^2 \sim
\frac{1}{2} (e^2 - 5) \log  ( \gamma_M /( 2 \pi ))
\eeq
as $M \to \In$.
Since $c(1) =1$, this means that on average $Z(t)^2$ at its maxima is
$1+ \frac{1}{2} (e^2 -7) = 1.1945 \dd$ times the average of $Z(t)^2$ over the entire range $0 < t \le \gamma_M$.
(This surprisingly small factor of 1.1945...
occurs because
the values of $Z(t)^2$ at the critical points where they achieve
their maxima are not weighted by the lengths of the intervals on which the maxima are computed.
Large values of $Z(t)$ are usually associated to large gaps between consecutive
zeros.)
Computation over the range from $T= \gamma_{n_2}$ to $T+H = \gamma_{n_3}$ yielded a value of 1.224... instead of the asymptotic value of
$1.1945 \dd $.
(The value 1.224... is probably a slight underestimate of the correct ratio,
since the actual maxima were not
determined, but the largest of the values at the 40 evenly spaced points was used.)

Gonek \cite{Gon1} has shown, again assuming the RH, that
\beql{eq296}
\frac{1}{M} \sum_{m=1}^M Z( \gamma_m + i \alpha \Delta )^2  \sim
\lt 1 - \lt \frac{\sin \pi \alpha}{\pi \alpha} \rt^2 \rt \log ( \gamma_M / ( 2 \pi ))
\eeq
as $M \to \In$,
where $\Delta = 2 \pi ( \log ( \gamma_M /( 2 \pi )))^{-1}$.
Computations for $\alpha =0.1$, $0.2$, $\dd , 0.9$ and over the zeros
numbered $n_4$, $n_4 +1 , \dd, n_5 -1$ showed reasonably
good argument, but
with the ratio of empirical data to Gonek's asymptotic estimate
declining by 4\% as $\alpha$ goes from 0.1 to 0.9.
%\section{Distribution of values of $\mbox{\protect\boldmath $\zeta$}$ {\bf ( 1/2 + it)}}
\section{Distribution of values of $\zeta ( 1/2 + it)$}\label{sc210}
\hspace*{\parindent}
Since
$$
\log \zeta (1/2 + it) = \log  | Z (t) | + \pi i S(t) ~,$$
it is not surprising that methods that yield the distribution of $S(t)$
should give corresponding results for $\log  | Z(t) | $.
Selberg in unpublished manuscripts studied
mean values of $( \log  \zeta ( 1/2 + it))^h ( \log \zeta (1/2 - it ))^k$ for
nonnegative integers $h$ and $k$, and his results imply, for example, that
for rectangles $E$ in $R^2$
\beql{eq2101}
\lim_{T \to \In}  \frac{1}{T} \left | 
\left \{ t :~ T \le t \le 2T ,
\frac{\log  \zeta (1/2 + it )}{( 2^{-1} \log \log T)^{1/2}} \in E \right \} \right |
= ( 2 \pi )^{-1} \iint_E
e^{- ( x^2 + y^2 )/2} dx dy~,
\eeq
so that in particular, for any $\alpha < \beta$,
\beql{eq2102}
\lim_{T \to \In}  \frac{1}{T} \left | 
\left \{
t :~ T \le t \le 2T, ~\alpha < \frac{\log |Z(t)|}{( 2^{-1} \log \log T )^{1/2}} < \beta \right \} \right | =
(2 \pi )^{-1/2}~\int_\alpha^\beta e^{- x^2 /2} dx ~.
\eeq
Thus the real and imaginary parts of $\log \zeta (1/2 + it )$ behave like independent
normal variables with means 0 and variances $( \log  \log  t) /2$.
While Selberg's results have
not been published,
they were known to some mathematicians (see \cite{Hej6,Joy1,Jut,Mon6},
and some extensions of Selberg's results have been obtained
by Joyner \cite{Joy1} and Tsang \cite{Ts2}.
The weaker result \eqn{eq2102} has been reproved by
Laurinchikas \cite{Lau1,Lau2,Lau3,Lau4,Lau5}.

The critical issue here is whether the approximation \eqn{eq2102} is accurate even for $T$ fixed and
$\alpha$ and $\beta$ varying over wide ranges.
If that is the case, then we are led to expect that something like \eqn{eq291}
holds.
Furthermore, if the approximation is accurate even for $\alpha$ and $\beta$ relatively
large (compared to $T$), one would expect that the maximal size of $|Z(t)| $, for $0 \le t \le T$, would be
$\exp (( \log  T)^{1/2 + o(1)} )$,
which is conjectured by some to be the true rate of growth of $Z(t)$
(cf.~Section~\ref{sc28}).
Thus it is of substantial interest to find out more about the tails of the distribution
of $\log |Z(t)| $.

For $n_0 \le n \le n_1 -1$,
$n_0 = 10^{12} - 6032$, $n_1 = n_0 + 10^6$,
each interval $( \gamma_n , \gamma_{n+1} )$ was partitioned into 40 equal
subintervals, $Z(t)$ was evaluated at the endpoints of these subintervals, and
a linear approximation to $Z(t)$ between consecutive evaluation points
was used to estimate
\beql{eq2103}
b_{\alpha , \beta} =
\frac{1}{\gamma_{n_1} - \gamma_{n_0}}  \left |\{ t :
\gamma_{n_0} \le t \le \gamma_{n_1} , 
\alpha \le \log | Z(t) | \le \beta \} \right |
\eeq
for $\beta = \alpha + 1/100$,
$\alpha = k/100$,
$- 1000 \le k < 1000$.
The mean of this distribution
(as derived from the
$b_{\alpha , \beta}$ data) was $5.29 \times 10^-4$ and the variance
was 2.2930.
(In Fig.~2.11.1 it is labelled as the $10 = 10^{12}$ distribution.)
Similar data was obtained for $n_2 \le n \le n_3 -1$, $n_2 = 10^{20} + 15, 316, 087$,
$n_3 = n_2 + 10^6$, and there the mean
was $5.20 \times 10^-4$ and the variance was 2.5657.
(This is the $N=10^{20}$ distribution.)
Based on \eqn{eq2102}, one would expect mean values of 0, which is very close to the
calculated values, given the errors in the computation
and sampling errors.
The values for the variances would be expected to be $( \log  \log  T) /2$,
where $T$ is the height of the data set,
which equals 1.635 and 1.894 for the two data sets,
respectively.
Since $( \log  \log  T) /2$ is only the asymptotic value and
increases very slowly, lower order terms can be expected to be significant, and so the agreement between observed data and theory is reasonably good on this point
as well.
However, the shapes of the observed distributions of $\log  |Z(t)| $ appear to be different
from the asymptotic normal distribution.
To obtain a good comparison, the two distributions for $N=11^{12}$ and $10^{20}$ were each scaled to have variance~1, and were plotted in Fig.~2.10.1 together with the standard normal distribution.
We see that while the fit of the $N=10^{20}$ data is slightly better than
that for $N=10^{12}$, it is not much better.
This is in great contrast to the fit of the data for $S (t)$ (which, apart from a factor of $1/ pi$, is the imaginary
part of $\log \zeta (1/2 +it)$, while
$\log  | Z(t)| $ is the real part of it) which, as we see in Section~\ref{sc24} and Fig.~2.5.1, is much better.
It might be of some interest to compute second order terms in the expansion of moments of $\log  | Z(t) | $ to see what is responsible for the deviations from the asymptotic behavior that are visible in the data.
In view of Goldston's results \cite{Go2} (mentioned in Sections~\ref{sc24} and \ref{sc26}), it seems likely that such higher order terms depend on the pair correlation of zeros, and even
on higher order correlations.

The area between the empirical distribution curve for $N=10^{12}$ in Fig.~2.11.1
and the normal curve is 0.132, while for $N=10^{20}$ the corresponding area is 0.114.
In both cases these areas are much larger than those for the distribution
curves for $S(t)$ discussed in Section~\ref{sc24}, which confirms the
impression one obtains by comparing Fig.~2.5.1 to Fig.~2.11.1.

Table~2.11.1 presents extensive data on the moments of $\log |Z(t)| $.
The six sets of data summarized in this table were all obtained by choosing
$10^6$ random points in an interval of length $1.5 \times 10^5$.
For $N=10^{12}$, this interval started near zero number $10^{12} - 6, 032$.
For $N=10^{18} (a)$ and $N=10^{18} (b)$, the intervals were the same,
starting near zero number $10^{18} - 8, 839$ but the random sequences were different, since different seeds were chosen for the
pseudorandom number generator.
This was done to estimate the size of the sampling error.
For $N=10^{20} (c)$, the starting point was near zero number
$10^{20} - 48, 776$, while for $N=10^{20} (d)$, it was near zero number
$10^{20} + 15, 316, 087$.
The mean and second moment for each data set are shown in the $k=1^{\ast}$ and $k=2^{\ast}$ entries,
respectively.
These were then applied to translate and scale the data sets to obtain mean equal to 0 and variance equal to 1,
for ease of comparison with the standard normal distribution.
The $k$-th entry in the table, $1 \le k \le 10$, given the $k$-th moment of
each scaled data set, and the last column gives the corresponding value for the normal
distribution (0 for $k$ odd, $(k-1) \cdot (k-3) \cdot \dd \cdot 3 \cdot 1$ for $k$ even).

Given that the distribution of $\log |Z(t)| $ differs so much from the
expected normal one, we have to treat the data about moments of $|Z(t)| $, for example,
with extreme caution, as they may not be representative of the true asymptotic behavior.
Furthermore, the general distribution of $Z(t)$ may be even
further from
what happens higher up.

Figure~2.11.2 presents some empirical data
on values of $Z(t)$.
This figure is based on the values of $Z(t)$ in the three intervals covering $2.8 \times 10^6$ zeros that were described in the preceding
section.
For each interval between consecutive zeros, the function $|Z(t)| $ was approximated
on 40 equal-sized subintervals by a linear function,
and the length of the interval on which this linear approximation was in each range
$[k-1, k)$ for $k \ge 0$ was computed.
If $A_k$ denotes the length of all the intervals on which the linear
approximations were in $[k-1, k)$, and
$$q_k = \frac{A_k}{\sum_{j=1}^\In A_j}
$$
the fraction of time spent there, then the plot in Fig.~2.11.2 shows $\log q_k$.
From this graph and other graphs based on the data from each of the
three main intervals separately, it appears that for $k <\sim 150$,
the empirical data obtained so far represents well the long-run statistics
of $|Z(t)| $
at the heights that were investigated.
In particular, the curves of $\log q_k$ for the three main intervals
are almost identical in that range.
On the other hand, for $k >\sim 250$, the behavior of $\log q_k$ is dominated
by a few large peaks of $|Z(t)| $ (which also account for a large part of the values
of high moments of $Z(t)$ dealt with in the previous section).
In particular, the segments of the graph in Fig.~2.11.2 that shoot up are caused by
high peaks.
The final region $(k \ge 353)$ is due to two peaks
where $|Z(t)| $ reaches
about
460, and the preceding
region of increase in $\log q_k$ is due to a point where $|Z(t)| $ is around 351.
%\section{Values of $\mbox{\protect\boldmath $\zeta '$} {\bf (1/2 + i} \mbox{\protect\boldmath $\gamma$} {\bf )}$}
\section{Values of $\zeta' (1/2 + i \gamma )$}\label{sc211}
\hspace*{\parindent}
Under the assumption of the RH and of a weak consequence of the pair correlation conjecture, namely that for some $\tau > 0$, there is a constant $B$ such that
\beql{eq2111}
\limsup_{N \to \In} \frac{1}{N} \Biggl|
\{ n : N \le n 2 N ,~ \delta_n < c \} \Biggr| \le Bc^\tau
\eeq
holds uniformly for all $c \in (0, 1)$, Hejhal \cite{Hej6}
has shown that for all $\alpha < \beta$,
\beql{eq2112}
\lim_{N \to \In} \frac{1}{N} \left | \left \{
n : N \le n \le 2N ,~ \frac{\log \left | \frac{2 \pi Z' ( \gamma_n )}{\log ( \gamma_n ( 2 \pi )^{-1} )} \right |}
{( 2^{-1} \log \log N)^{1/2}} \in ( \alpha , \beta ) \right \} \right| =
( 2 \pi )^{-1/2} \int_{\alpha}^\beta
e^{- x^2 / 2} dx ~.
\eeq
(Note that under the RH, which we assume throughout this section,
$| \zeta' ( \rho ) | = |Z' ( \gamma )| $ if $\rho = 1/2 + i \gamma$.)

As was the case with the values
of $Z(t)$, we would like to obtain more information about the tails of the distribution of
$Z' ( \gamma_n )$, and in particular about the moments.
Let us define
\beql{eq2113}
J_\lambda (T) = \sum_{\gamma_n \le T} ~| Z' ( \gamma_n ) |^{2 \lambda} ~.
\eeq
Then $J_\lambda (T)$ exists for all $\lambda \ge 0$, and if the zeros of the zeta function are all simple (as they are conjectured to be, and as
is the case with all of the zeros that have been dealt with numerically) then $J_\lambda (T)$ also exists for $\lambda < 0$.
The only nontrivial
asymptotic result
was proved by
Gonek \cite{Gon1} under the
assumption of the RH;
$$J_1 (T) \sim \frac{T}{24 \pi}  ( \log  T)^4 ~~~\mbox{as} ~~~
T \to \In ~.
$$
It is clear that $J_0 (T) = N(T) \sim \frac{T}{2 \pi} \log T$ as
$T \to \In$, and it is known
(cf.~\cite[Section~14.27]{Tit2}) that
$J_{- 1/2} (T) /T \to \In$ as $T \to \In$.
Gonek \cite{Gon3} has also shown that (under the RH)
$J_{-1} (T) \ge cT$ for some $c > 0$.
If the limit law \eqn{eq2112} holds even for small $N$,
and the tails of the distribution of $Z' ( \gamma_n )$ are not too large,
then we might expect (as was suggested by Hejhal \cite{Hej6} and stated explicitly
by Gonek \cite{Gon3}) that
$J_\lambda (T)$ is on the order of
\beql{eq2114}
T( \log T )^{( \lambda +1)^2} ~~~\mbox{as} ~~~ T \to \In ~.
\eeq
Furthermore, Gonek \cite{Gon3} has conjectured that
\beql{eq2115}
J_{-1} (T) \sim 3 \pi^{-2} T ~~~\mbox{as} ~~~T \to \In ~.
\eeq

Approximate values were obtained for $|Z' ( \gamma_n )| $,
$n_0 \le n \le n_0 + 10^6 -1$, where
$n_0 = 10^{20} + 15, 316, 107$.
Since the behavior of $Z(t)$ is determined primarily by zeros close to $t$
(cf.~\cite{BH,Hej6}),
it was assumed that for $t$ near $\gamma_n$, $Z(t)$ is approximated well by
\beql{eq2116}
a \prod_{j=-20}^{20}  (t- \gamma_n+j ) ~,
\eeq
where $a$, representing the influence of zeros further from $\gamma_n$, is almost a constant, and this led to approximating $Z' ( \gamma_n )$ by
\beql{eq2117}
\epsilon^{-1} Z( \gamma_n + \epsilon )
\prod_{j=-20 \atop j \neq 0}^{20}
\frac{\gamma_n+j}{\gamma_n + \epsilon - \gamma_n+j} ~,
\eeq
where $\epsilon = ( \gamma_{n+1} - \gamma_n ) /40$.
Varying the number of terms in the heuristic approximation
\eqn{eq2116} as well as varying $\epsilon$ showed that \eqn{eq2117} does produce
good approximation to $Z' ( \gamma_n )$.

The smallest value of $| Z' ( \gamma_n )| $ that was found was 0.13,
while the largest was $2.47 \times 10^3$.
The values of $\log  |Z' ( \gamma_n )| $ had a mean of 3.35 and a variance of 1.14,
in contrast
to 1.91 and 1.9, respectively, which are
predicted by Hejhal's result \eqn{eq2112}.
Given the slow rate of growth of these quantities, second order terms
in the asymptotic results are likely to be large, so this difference between expected and observed values is probably not significant.
If we let
\beql{eq2118}
v_n = ( \log  | Z' ( \gamma_n )| - m ) / \sigma ~,
\eeq
where $m$ is the mean and $\sigma$ the standard deviation of our set of $\log  |Z' ( \gamma_n )| $, then Fig.~2.12.1 shows a
comparison of the distribution of $v_n$ with the standard normal distribution.
The line is the standard normal density,
while the scatterplot represents a histogram of $v_n$;
for each interval $[ \alpha , \beta )$, $\alpha = k/50$, $\beta = \alpha + 1/50$,
a star is placed at $(x, y)$, $x = (( \alpha + \beta ) /2-m)/ \sigma$,
$y = \sigma  b_{\alpha , \beta}$, where
\beql{eq2119}
b_{\alpha , \beta} = {50}{10^6} \left |  \{ n :~ v_n \in [ \alpha , \beta ) \} \right | .
\eeq
It is worth noting that the distributions of $\log  | Z (t)| $ and $\log  |Z' ( \gamma_n ) | $ are both supposed to be asymptotically normal,
but the convergence appears much faster for $\log  | Z' ( \gamma_n )| $,
as is revealed by a comparison of Fig.~2.12.1 to Fig.~2.11.1.
This is true even though the asymptotic normality of $\log  |Z(t)| $ is an unconditional
theorem, while that of $\log |Z' ( \gamma_n ) | $ depends on unproved assumptions.

Table~2.12.1 lists the moments of the $v_n$ and of the asymptotic normal distribution.
The entry for $k=1 , \dd , 10$ denotes the $k$-th
moment of $v_n$ and the normal distribution, while the $k=1^{\ast}$ and $k=2^{\ast}$ entries give the first two moments of $\log  |Z' ( \gamma_n ) | $,
respectively.
Comparison with Table~2.11.1 again shows much better agreement between empirical and expected values
for $\log |Z' ( \gamma_n ) | $ than for $\log |Z(t)| $.

Moments of $Z' ( \gamma_n )$ for the $10^6$ values that
were computed are shown in Table~2.12.2.
Since \eqn{eq2114} suggests that for $M$ relatively large,
\beql{eq21110}
J_\lambda^{\ast} (M, N) = \frac{1}{M} \sum_{n=N+1}^{N+M} 
| Z' ( \gamma_n ) |^{2 \lambda}
\eeq
ought to be of the order of magnitude of
$$
( \log  \gamma_N )^{( \lambda +1)^2 -1} ~,
$$
while for $\lambda =1$ and $\lambda = -1$ we ought to have the more precise relations
\begin{eqnarray}
\label{eq21111}
J_1^{\ast} (M, N) & \sim & \frac{1}{12} ( \log  \gamma_N )^3~, \\
\label{eq21112}
J_1^{\ast} (M, N) & \sim & 6 \pi^{-2} ( \log \gamma_N)^{-1}
\end{eqnarray}
(the asymptotic relations holding as $M, N \to \In$ with $M$ relatively large).
Table~2.12.2 shows the ratio of empirical to
expected values, namely
\beql{eq21113}
r_\lambda = J_\lambda^{\ast} ( 10^6 , n_0 -1 ) ( \log  \gamma_{n_0} )^{1- ( \lambda +1)^2} ~.
\eeq

The value for $\lambda =1$ is in excellent agreement with \eqn{eq21111} (which is a theorem under the assumption of the RH),
while the value for $\lambda =-1$ is reasonably consistent with that of
\eqn{eq21112}.
Since \eqn{eq21112}
is derived from Gonek's conjecture \eqn{eq2115}, this
supports the conjecture.

A theorem announced by Fujii \cite{Fu4} (which assumes the RH) states that
\beql{eq21114}
\begin{array}{r@{~}l@{~}l}
\ds_{0 < \gamma_n \le T} \zeta' ( 1/2 + i \gamma_n ) & = &
\df{T}{4 \pi}  \log^2  \df{T}{2 \pi} +
c_0  \df{T}{2 \pi}  \log \df{T}{2 \pi} \\
~~ \\
&& + c_1 ~ \df{T}{2 \pi} + O(T^{9/10 + o(1)} )
\end{array}
\eeq
as $T \to \In$, where $c_0$ and $c_1$ are explicit constants.
This turns out to be in excellent agreement with the empirical result
\beql{eq21115}
\sum_{n=n_0}^{n_0 +10^6 -1} \zeta' (1/2 + i \gamma_n ) =
2~.181 \times 10^6 + i 8.7 \times 10^3 .
\eeq

The approximate procedure \eqn{eq2117} that was used to estimate $Z' ( \gamma_n )$ can be replaced by a much more
rigorous and accurate method.
The algorithm of
cite{OS} that was used to compute $Z(t)$ precomputes a set of values from which $Z(t)$ is obtained
by interpolation.
However, the main interpolation formula \eqn{eq4315} can be differentiated
with respect to $t$, which enables one to compute
$Z' (t)$ (and therefore also $\zeta' (1/2 + it )$) from the
basic data.
If such a program were written, it could be used to check the speed of
convergence of the distribution of
$\log  | \zeta' (1/2+ it)| $ to the Gaussian
limit that has been shown to hold under the assumption of the RH by Hejhal \cite{Hej6}.
\section{Gram points and blocks}\label{sc212}
\hspace*{\parindent}
{\em Gram's law} is the empirical observation that
$Z(t)$ usually changes sign in each
{\em Gram interval}
$G_n = [g_n , g_{n+1} )$, $n \ge -1$.
(The Gram points $g_n$ are defined in Section~\ref{20}.)
Gram \cite{Gram} observed that it held in the range of values he
investigated, but he conjectured that it would fail eventually.
The first counterexample occurs for $G_125$, and was discovered by Hutchinson \cite{Hu}.
If Gram's law held universally, the RH would obviously be true.
However, it is known that this ``law'' fails infinitely often.
On the other hand, it does hold for a large fraction of cases.
For $n \le 1.5 \times 10^9$, Gram's law holds 72.79\% of the time \cite{LRW2}, among $10^6$ Gram intervals near zero number $10^{12}$, it holds
70.82\% of the time, and among $10^6$ Gram intervals near zero number $10^{20}$, it holds 68.9\% of the time.
(Under the GUE and some further assumptions to be discussed later, one might expect
that asymptotically, Gram's law should hold about 66.3\% of the time.)

One barely plausible reason Gram's law might hold (and why the RH might hold) is that in the Riemann Siegel formula for $Z(t)$ (see Eq.~\eqn{eq412}) the
leading term equals $2(-1)^n$ at $t=g_n$.
If this term, which is the largest, were always dominant, then Gram's law and the RH would follow.
We now know this to be false, but there is still interest in the behavior
of $Z(t)$ at Gram points, since sign changes of $Z(t)$ correspond to zeros of the zeta function on the critical
line.

A Gram point $g_n$ is called
{\em good} if
$(-1)^n Z(g_n ) > 0$, and
{\em bad}
otherwise.
A {\em Gram block} is an interval
$B_n = [g_n , g_{n+k} )$ such that $g_n$ and $g_{n+k}$
are good Gram points, while $g_{n+1} , \dd , g_{n+k-1}$ are bad Gram points.
The {\em length} of a Gram block
$B_n = [g_n , g_{n+k} )$ is $k$.
The {\em pattern of zeros} in a Gram block $B_n = [g_n , g_{n+k} )$ is the string $a_1 \cdots a_k$, where $a_i$ denotes the number of zeros of
$Z(t)$ in $[g_{n+i-1} , g_{n+i} )$.
Since no Gram interval with more than 4 zeros has even been found,
writing $a_1 \cdots a_k$ without comma separators is unambiguous.
(Gram intervals with arbitrarily many zeros almost surely exist, but given the GUE predictions about zeros repelling each other, they
are likely to be rare.)

The statistics that have been collected on Gram intervals and blocks (as well as on exceptions to Rosser's rule, which are discussed in Section~\ref{213}) are subject to
errors, not only because of the roundoff problems that have been 
mentioned before and are discussed extensively in Section~\ref{sc4}, but also
because
even if the
computations of $Z(t)$ were exact,
Gram points were determined only approximately,
so that the determinations of the signs of $Z(g_n )$ were not always certain.
No special precautions were taken to deal with this problem
(such as checking on the size of the computed value of $Z(g_n )$) as it was felt that this was unlikely to affect general statistics.

The computations of \cite{LRW2} of the first $1.5 \times 10^9$ zeros found only 6 Gram
blocks of length 9, and none of lengths $\ge 10$.
In contrast, the maximal lengths of Gram blocks found during the present
computations were 9 for $N=10^{12}$,
9 for $N=10^{14}$, 11 for $N=10^{16}$, 13 for $N=10^{18}$ (1 case), 12 for $N=10^{19}$, 14 for $N=10^{20}$ (2 cases, with zero patterns $01111111113110$ and $01111111111130$),
and 13 for $N = 2 \times 10^{20}$.

Table~2.13.1 gives the fraction of Gram blocks in given data sets with given
lengths.
The $N=1$ and $N=1.4 \times 10^9$ data is derived from
Table~1 of \cite{LRW2},
and comes from two sets of $10^8$ Gram intervals each, the first one starting
at $g_0$, the second at $g_n$ for $n=1.4 \times 10^8$.
The $N=10^{12}$ data is based on only 1,590,000 Gram intervals.

The main program did not keep track of Gram blocks according to their pattern of zeros.
However, a special study was made of two blocks of $10^6$ Gram intervals each,
one starting at $g_{n_1}$,
$n_1 = 10^{12} - 6, 034$,
the other at $g_{n_2}$,
$n_2 = 10^{20} - 42, 780$,
which for the remainder of this section will be
referred to as the $N=10^{12}$ and $N=10^{20}$ data sets, respectively.

If a Gram block $B(n, k)$ contains exactly $k$ zeros (so is not associated with a violation of Rosser's rule, see Section~\ref{sc213}) then its zero pattern must be either $211...110$,
or $011...112$, or $011...131...110$ (where
the data~... refer to any strings of consecutive 1's, but these strings might be
even shorter than indicated).
pattern might be missing).
Van~de Lune
et~al. \cite{LRW2}
noted
in their computations that for a fixed
$k$, the first two zero patterns seemed to be much more frequent than the third one,
and that the frequencies seemed stable as the height of zeros increased.
The new computations, however, show a steady decrease in the frequency with which the third pattern
appears
as the height increases.
Table~2.13.2 shows the
observed frequencies.
The $N=1$ entry is drawn from Table~2 of \cite{LRW2}, which is based on statistics of 3 sets of $10^8$
Gram intervals each, starting at
$g_m$ with $m=0$, $7 \times 10^8$, and $1.4 \times 10^9$,
respectively.
Only
Gram blocks of length $k$ with exactly
$k$ zeros are considered, and the entry in the table gives the fraction
of all such Gram blocks that have a zero pattern
containing a 3.
The decrease in the frequency of the third zero pattern is puzzling.
The GUE theories suggest that this pattern ought to occur a positive proportion of the time.

Table~2.13.3 presents data on the fraction of Gram intervals that contain a given number
of zeros.
The $N=1$ and $N= 1.4 \times 10^9$ data sets are the same as in Table~2.13.1,
and these entries come from Table~5 of \cite{LRW2}.
Note that there were no Gram intervals with $\ge 4$ zeros in the $N=10^{12}$ and $N=10^{20}$ sets (although such intervals did turn up in other data sets around the $10^{20}$-th zero, for example).

The GUE entry in Table~2.13.3 was derived by assuming that a Gram interval does not differ
from any other interval of that length, and so the entry in the table for a given $m$ in the GUE row is the probability that an interval of length 1 contains exactly $m$ zeros.
Since the averages of $S(t)$ do increase as $t$ increases, it seems reasonable
to expect that at large heights the local distribution of the zeros will be independent
of Gram points, which leads to the above assumption (cf.~\cite{Fu4}).
In other words, the expectation is that at large heights, any grid
of points spaced like the Gram points would exhibit similar
behavior with respect to location of zeros.

If the zeros at large heights are distributed independently of the Gram points,
in the sense above, namely that shifting all the Gram points in a large
interval by the same amount would not affect the statistics of Gram intervals and blocks, then we can expect that if we define
$$z_n = \frac{\gamma_n - g_m}{g_m+1 - g_m} ~~~~\mbox{if} ~~~~
\gamma_n \in [g_m ~, g_{m+1} ) ,
$$
then the $z_n$ will be distributed uniformly in the unit interval \cite{Fu4}.
(So far, the equidistribution of the $\gamma_n$ has been shown only modulo much coarser grids,
see \cite{Hl,Fu3}.)
Figure~2.13.1 shows the distribution of $z_n$ for the two data sets $N=10^{12}$ and $N=10^{20}$
as well as for a third set, labelled $N= 10^6$, derived from the $10^6$ zeros $\gamma_n$ with
$10^6 + 1 \le n \le 2 \times 10^6$.
In each case a histogram was prepared giving the number of $z_n \in [j/1000$,
$(j+1)/1000)$, $0 \le j < 10^3$, and these data were used to derive the smooth curve in the picture
using the lowess function of \cite{BC}.
A perfectly uniform distribution would correspond to a straight horizontal segment at height~1,
while the most nonuniform distribution (which also would minimize the moments of $|S(t)| $), corresponds to a point mass at 1/2 and 0 elsewhere.
The $N=10^{20}$ curve is much closer to this conjectured uniform behavior than the $N=10^{12}$ curve,
and neither is far away from it.
Even the $N=10^6$ curve is not very far from uniform behavior.
The area between the curve in Fig.~2.13.1 and the straight horizontal segment at height 1 is 0.105 for $N=10^6$, 0.051 for $N=10^{12}$, and 0.028 for $N=10^{20}$.
Thus is appears to be of the order of $( \log N)^{-1}$.

A quantitative study of the extent to which the sign of $Z(g_n )$ might coincide with $(-1)^n$ was started by Titchmarsh \cite{Tit0}, who showed (as might be
expected from the Riemann-Siegel formula \eqn{eq412}) that as $M \to \In$,
\begin{eqnarray}
\label{eq2121}
M^{-1} \sum_{n=1}^M Z(g_n) & = & o(1) ~, \\
M^{-1} \sum_{n=1}^M (-1)^n Z(g_n) & = & 2 + o(1) ~,
\label{eq2122}
\end{eqnarray}
as well as
\beql{eq2123}
M^{-1} \sum_{n=1}^M ~ Z(g_n ) Z(g_{n+1} ) = - 2(1+c_0 ) + o(1)
\eeq
where $c_0 = 0.577 \dd$ is Euler's constant.
These results have been strengthened and extended considerably by Moser \cite{Mos1,Mos2,Mos3,Mos4,Mos5,Mos6,Mos7,Mos8,Mos9,Mos11,Mos12,Mos14}.
Table~2.13.4 presents some averages involving the $Z(g_n )$ that were computed,
using the 2 sets of $10^6$ values each that were specified above.
For example, the $|Z^3 (g_n ) | $ entry gives the value of
$$10^{-6} \sum_{n=R}^{R+ 10^6 -1}  |Z^3 (g_n ) |
$$
for the appropriate $R$.
We see that the computational results are in excellent
agreement with Titchmarsh's results \eqn{eq2121} to \eqn{eq2123}.
\section{Violations of Rosser's rule}\label{sc213}
\hspace*{\parindent}
Rosser's rule, formulated on the basis of empirical evidence, states that a Gram block $B(n, k)$ contains
at least $k$ zeros.
It thus requires less regularity than Gram's law, yet if Rosser's rule were to hold universally, it would imply the RH (just as the validity of Gram's law would), and would also imply that every Gram block $B(n, k)$ contains exactly $k$ zeros.
However, it is easy to see that Rosser's rule holding up to height $T$ is equivalent
to the bound $|S(t)| < 2$ holding for $t < T$, which contradicts the unboundedness of $S(t)$.
Thus Rosser's rule has to fail infinitely often.

$S(t)$ grows very slowly, and Rosser's rule holds for most Gram blocks that have been checked.
The first exception to Rosser's rule (defined as a Gram block $B(n, k)$ which has fewer than $k$ zeros)
is $B(n, 2)$ with $n = 13, 999, 525$
\cite{Br5}.
There are only 15 exceptions to Rosser's rule for $n \le 7.5 \times 10^7$ \cite{Br5}, and 3055 exceptions for $n \le 1.5 \times 10^9$ \cite{LRW2}.
Among the values of $n$ with $1.4 \times 10^9 \le n \le 1.5 \times 10^9$,
there were 0.287 exceptions per $10^6$ zeros.

The new computations found 62528 exceptions to Rosser's rule.
Table~2.14.1 shows how many occurred in each data set and their density.
Not only are the exceptions in the new data sets more frequent, but they also
are much more varied than those found among
the first $1.5 \times 10^9$ zeros.
If $B(n, k)$ is an exception to Rosser's rule, then $k$ will be referred to as the
{\em length} of the exception.
The pattern of zeros inside this block has to be $011 \cdots 110$.
(For notation, see Section~\ref{sc212}.)
To describe the exception, we have to specify where the two ``missing zeros'' are located.
We will use the notation
\beql{eq2131}
kX a_1 a_2 \cdots a_m , ~~~~X = L ~~\mbox{or} ~~ R ~,
\eeq
to denote an exception $B(n, k)$ where the missing zeros are to the left of
$B(n, k)$ (if $X=L$) or to the right of
it (if $X=R$), and where $a_1 a_2 \cdots a_m$ denotes the pattern of zeros in the smallest union of Gram blocks that is adjacent to $B(n, k)$ and contains the missing zeros.
Thus, for example, $3L0312$ denotes an exception of length 3, where the pattern of zeros in $[g_n-4 , g_n+3 )$ is 0312010.
This is not a completely unambiguous description, but it suffices
for all the cases that have been encountered, as no case of 3 exceptions to
Rosser's rule that are close together has been found.
We will refer to \eqn{eq2131} as the
{\em type} of the exception, and $m$ will be called the
{\em length of the excess block}.
With this notation the 3055 exceptions among the first $1.5 \times 10^9$ zeros fall into just 13 types:
$$
\begin{array}{l}
2R3 ,~~2L3 ,~~2R40, ~~2L04, ~~2R22 ,~~ 2L22 ,~~2R230~, \\
~~ \\
2L032, ~~2R410 ,~~3R3, ~~3L3, ~~3R40 ,~~3L04 ~,
\end{array}
$$
with 2715 of them being of types $2R3$ and $2L3$, and only 82 being of length 3.
In particular, all lengths of exceptions and lengths of excess blocks are $\le 3$.

The exceptions found during the new computations
fall into 206 distinct types.
The relative frequencies of the most popular types in the new data sets and also among the first $1.5 \times 10^9$ zeros
are shown in Table~2.14.2.

The maximal length of an exception that was found is 9, and it occurs in two
exceptions, one of type $9R3$
and one of type $9L3$.
There are 15 exceptions of length 8, 84 of length 7, and 416 of length 6.
The maximal length of an excess block is 9, and occurs in 2 exceptions of types
$2R211113110$ and
$3L011311112$.
There are 8 cases of excess blocks of length 8, and 46 of length 7.

Some of the 62528 exceptions to Rosser's rule that were found in the main computations occur very close to each other.
There are several cases where 2 exceptions are separated by a single Gram interval.
The smallest such case that was found is that of $B(n, 3)$ and $B(n+4, 5)$, where
$n=10^{16} + 3, 916, 331$, and the pattern of zeros in $[g_n , g_{n+1}0 )$ is
0103011103.
No case was found where two exceptions are adjacent.
(However, Section~\ref{sc3} presents results of other computations that found several examples
of this phenomenon.)
Finally, no example of 3 exceptions close to each other has been found.

%  FILE: ch3.tex

\chapter{Special points for the zeta function}\label{sc3}
\section{Introduction}\label{sc30}
\hspace*{\parindent}
The main computations described in Section~\ref{sc2} were carried out at heights that were thought likely to be random
as far as the behavior of the zeta function is concerned.
Thus the data that
were
collected
were
likely to
represent
long-run statistics of the zeta function at
those heights.
However, what is most interesting is not to study the typical
behavior but to look at extreme values.
It would be desirable, for example, to determine where the smallest spacing of consecutive zeros
up to some height is without finding all the zeros up to that height.
No way to do this is known.
It is not even known how to find places where consecutive zeros are very close to each other.
The problem is that we would need a way to determine places where both
the zeta function and its derivative are small, and this is not feasible
currently.
On the other hand, there are ways to determine values of $t$ where $\zeta (1/2 + it )$ is likely to be very
large.
Such methods have been used before \cite{KW,vdL,Od2},
and the method described later in this section is a development
of the method that was mentioned briefly in \cite{Od2}.
These methods determine values of $t$ for which the large initial terms in formulas
for $\zeta (1/2 + it )$ have the same argument, and therefore add up to a large
quantity that often is not cancelled by the remaining terms.

One reason for the interest in large values of
$\zeta (1/2 + it )$
is that one can think of a large peak as ``pushing aside'' the zeros that would normally lie in that area, and if these zeros were pushed off the critical
line, one would find a counterexample to the RH.
No such counterexamples were found in the computations described here, but many interesting
phenomena were observed.

Section~\ref{sc31} presents the results of the computations near the special points.
Section~\ref{sc32} describes the diophantine approximation algorithms that were used to construct these special points.
Finally, Section~\ref{sc33} discusses how these algorithms could be improved, and what other computations could be attempted in the future.
\section{Computational results}\label{sc31}
\hspace*{\parindent}
The computations of this section, which are summarized in Tables~3.2.1 and 3.2.2,
found 5,168,540 zeros.
As is the case with the main sets of zeros described in Section~\ref{sc2}, even if the programs are correct and roundoff errors do not matter, it is not absolutely certain that the few dozen
zeros at the ends of the data sets are indeed all the zeros in those ranges (cf.~Section~\ref{sc21}).
However, for the purpose of exposition, it will be assumed that they are.

There were 22 separate computations, and the sets of zeros and special points associated to them will be denoted by the letters A through V.
Table~3.2.1 shows the first zero of each data set, the number of zeros in that set,
and the value of $t_0 = ( \gamma_n + \gamma_{n+1} ) /2$ for that $n$ for which
$|Z(( \gamma_n + \gamma_{n+1} )/2)| $ is largest among all $n$ in the data set.
Table~3.2.2 then shows the value of $Z(t)$ at $t=t_0$, the largest
value of $S(t)$ in a neighborhood of $t_0$ (which is
always
the largest $S(t)$ in a given data set,
but which does not always occur at $\gamma_n$ or $\gamma_{n+1}$),
the value of $\delta_n$ (which in all cases is the largest $\delta_m$ in a given data set),
and the pattern of zeros in a union of Gram blocks that include $t_0$ (see
Section~\ref{sc212} for notation).

The entries in Table~3.2.2 show that the attempt to produce unusual behavior of the zeta function was successful.
The value of $|Z(t)| \approx 1580$ found in set U is far higher than 641, the largest value that was found in
the main computations.
Similarly, the value of $\delta_n = 5.1454$ from set C is the
largest $\delta_n$ that has been found so far, and the value of
$S(t) = 2.8747$ from set T is a record for this function.
Figures~2.2.1 and 3.3.2 show graphs of $Z(t)$ near the special value of $t=t_0$ from set T,
and Fig.~3.2.3 shows a graph of $S(t)$ in that same range.

Figures~3.2.1 and 3.2.2 are typical of those for the other sets in that they display
a single high peak of $|Z(t)| $, with other nearby values of $Z(t)$ much smaller.
For example, in looking at stretches of about 30 Gram intervals centered at the special points,
one finds only 3 peaks among all 22 data sets where the sign of $Z(t)$ was opposite to that at the main peak,
and $|Z(t)| > 30$ was satisfied.
The largest value of $|Z(t)| $ in such secondary peaks was 36.
Thus we are probably still not seeing the expected behavior of large values of
$Z(t)$ that is discussed at the end of Section~\ref{sc28}.

The general distribution of zeros as well as other properties of the zeta function
in the ranges covered here were not too remarkable, aside from the behavior
near the peaks of $|Z(t)| $.
Exactly 3 midpoint values $w_n$ (see \eqn{eq272} for a definition) that were $> 250$ were found away from the
special values of $t$, but they were all $< 304$.
Exactly 100 values of $w_n < 5 \times 10^{-4}$ were found,
the smallest of them $2.47 \times 10^{-5}$.
The smallest value of $\delta_n$ that was found was $3.29 \times 10^{-3}$, with the
second smallest $7.62 \times 10^{-3}$.
(Since the probability the minimal $\delta_n$ of 5,168,540 being drawn
from the GUE ensemble
is
$\le 3.29 \times 10^{-3}$ is about 0.18,
this is consistent with the tendency that was observed
before of the minimal $\delta_n$ being somewhat smaller than expected.)
There were 5459 values of $n$ with $\delta_n  < 0.1$, and 844 values of $n$ with $\delta_n  > 2.8$.
The largest 22 $\delta_n$ that were found are the ones given in
Table~3.2.2.
The 23-rd largest $\delta_n$ was 3.50.
There were 1861 values of $\delta_n + \delta_n  < 0.6$, the smallest
of them 0.2512, and 525 values of $\delta_n + \delta_{n+1} > 4$,
the largest of them 6.0165.
(If $n=35, 200, 636, 070, 992, 305, 894$, so that
$\delta_n = 4.3214$ is the largest $\delta_m$ in set V, then
for this $n$ we have $\delta_{n-1} + \delta_n = 6.0165$.)

An initial concern about these computations was that they might give a distorted
view of various properties of the zeta function, such as the distribution of
$\delta_n$, for example, at the heights being investigated.
This was
thought possible because
the special points $t_0$ were chosen so that the
initial terms in the Riemann-Siegel formula for $Z(t_0 )$ behave as if $t_0$ were close to 0.
Thus it seemed possible that aside from the vicinity of the special point $t_0$,
where $Z(t)$ is large, $Z(t)$ might behave as if $t$ were small,
and so would be very constrained.
However, that appears not to be the case.
The agreement between the distributions of the $\delta_n$ in our small sets
and the GUE prediction is good when one compares graphs prepared like those
of Figures~2.4.4 and 2.4.6, and also when one prepares $q-q$ plots.
In those comparisons, the presence of one huge outlier does not make much of a difference.
On the other hand, when comparing moments of $\delta_n -1$, one sees substantial differences, especially for high moments.
These are easy to explain.
When computing the mean value of $( \delta_n -1 )^{10}$ over $2.5 \times 10^5$ zeros,
for example, a single value of $\delta_n = 5$ will contribute
$4^{10} \cdot 4 \cdot 10^{-6} = 4.1943$ to the mean, whereas the GUE prediction
for that mean is only 0.488.

The maximal value of
$|S(t)| $ does not always occur
at
one of the two zeros adjacent to the highest peak of $|Z(t)| $.
For example, in set V, if we let
$n= 35, 200, 636, 070, 992, 171, 653$, then $w_n =1329.5$,
$\delta_n = 4.3214$, but $\delta_{n-1} = 1.6951$,
and $S( \gamma_{n-1} + )  =  2.8314$,
$S( \gamma_n -)  =  1.1363$, $S( \gamma_n + ) = 2.1363$,
$S( \gamma_{n+1} - ) = 2.1851$, $S( \gamma_{n+1} + ) = -1.1851$.

Exactly 614 exceptions to Rosser's rule were found.
They fall into 45 types, each of which had occurred in the main computations.
The longest exception had length 6, and the longest excess block also had length 6.
On the other hand, a new phenomenon was observed in 21 of the 22 data sets, namely that of 2 exceptions to Rosser's rule being adjacent to each other.
Thus for example, the zero pattern 22000022 near the special point $t_0$ for set A
corresponds to an exception of type
$2L22$ followed immediately by an exception of type $2R22$.
This phenomenon has been observed only in the 21 cases exhibited in Table~3.2.2.

The basic conclusion to be drawn from the computations of this section is that the idea of looking for special points where the zeta function behaves in unusual ways is sound, and does produce interesting results.
It also shows that investigating only a random selection of about $10^8$ out of the first $10^{20}$ zeros misses some of the most intriguing places.
\section{Diophantine approximation algorithms and special points}\label{sc32}
\hspace*{\parindent}
The Riemann-Siegel formula (Eq.~\eqn{eq412}),
as well as other ``approximate functional equations'' show that the size of $\zeta (1/2 +it)$ is determined by the size of the sum of an initial segment of the divergent Dirichlet series
\beql{eq321}
\sum_{n=1}^\In  n^{-1/2 -it}  ~.
\eeq
One can also hope that the size of this sum is determined largely
by the size of a partial Euler product,
\beql{eq322}
P_X (t) = \prod_{p \le X}  (1-p^{-1/2 -it} )^{-1} ~.
\eeq
The basic strategy for finding large values of $\zeta (1/2 +it )$ is to find $t$ such that
$|P_X (t)| $ is large, and if it is, compute
$\zeta (1/2 +it )$.
(In practice, it has turned out to be helpful to first check that $| P_Y (t)| $ is large for some $Y > X$.
This eliminated many candidate values of $t$.)
There is no guarantee that this approach will succeed, but it appears to work well.

To find values of $t$ that make $|P_X (t)| $ large, we search for values of $t$ such that each of the $p^{it}$ is close to 1, as that makes each term in the product maximal.
Thus we need to find a $t$ for which there exist integers $m_1 , \dd , m_n$ such that each
of $t \log p_k - 2 \pi m_k$ is small,
$1 \le k \le n$, where $n= \pi (X)$ and $p_1 , p_2 , \dd , p_n$ are the
primes $\le X$.
This is an instance of a homogeneous simultaneous
diophantine approximation
problem.
We solve it using the Lov\'{a}sz
lattice basis reduction algorithm \cite{LLL}, which
has now become the basic tool in solving a variety of diophantine approximation problems in high dimensions.
Given a basis for a lattice in which the vectors have integer coordinates, this algorithm
produces another basis of short vectors.
While the new {\em reduced} basis is not guaranteed to contain the shortest
vector in the lattice, the algorithm has polynomial running time and variants
of it are efficient in practice.
The papers \cite{LO2,OtR} contain some examples of the applications of this algorithm.

The lattices to which the Lov\'{a}sz
algorithm was applied have as
their basis the rows of the following
$(n+1) \times (n+1)$ matrix:

\vsp
\beql{eq323}
\left(
\matrix{
[ \alpha_1 2^{m-r} \log p_1 ] & [ \alpha_2 2^{m-r} \log p_2 ] & \cdots & [ \alpha_n 2^{m-r} \log p_n ] & 1 \cr
[2 \pi \af_1 2^m] & 0 & \cdots & 0 & 0 \cr
0 & [2 \pi \alpha_2 2^m] & \cdots & 0 & 0 \cr
\vdots & ~ & ~ & ~ & ~ \cr
0 & 0 & \cdots & [2 \pi \alpha_n 2^m] & 0 \cr}
\right) ~,
\eeq

\vsp
\noindent
where $\alpha_k = p_k^{-1/4}$, and $m > r > 0$ are integers.
A typical vector in the reduced basis is then of the form
\beql{eq324}
(M [ \alpha_1 2^{m-r} \log p_1 ] - m_1 [ 2 \pi \alpha_1 2^m ] , \dd , M [ \alpha_n 2^{m-r} \log p_n ] - 
m_n [ 2 \pi \alpha_n 2^m ] ,~M)~,
\eeq
where $M ,  m_1 , \dd , m_n$ are integers.
For this vector to be short, $M$ and each of
\beql{eq325}
M [ \alpha_k 2^{m-r} \log p_k ] - m_k [ 2 \pi \alpha_k 2^m ]
\eeq
have to be relatively small.
For the difference in \eqn{eq325} not to be large,
\beql{eq326}
M2^{-r} \log p_k - 2 \pi  m_k
\eeq
must be small, so that $t = M 2^{-r} , m_1 , \dd , m_n$ gives a solution
to our basic problem.

The function of the $\alpha_k$ in the definition \eqn{eq323} of the lattice basis
is to take advantage of the fact that in trying to make $P_X (t)$ large,
it is more important that the $p=2$ term be large than that the $p=79$ term be large, say.
If
\beql{eq327}
t \log p_k - 2 \pi m_k = \epsilon_k~,
\eeq
and the $\epsilon_k$ are small, then
\beql{eq328}
\log  |P_X (t) | - \log  P_X (0) \approx - \frac{1}{2} \sum_{p \le X} 
\epsilon_k^2  p_k^{-1/2}  ~,
\eeq
and so we really wish to minimize $\sum \epsilon_k^2 p_k^{-1/2}$.
Since the Lov\'{a}sz algorithm attempts to
minimize the Euclidean norm of vectors, the definition of the $\alpha_k$ induces it to produce
the desired result.

The implementation of the Lov\'{a}sz algorithm that was used in the computations of this section was essentially the same as that of \cite{LO2,OtR},
and will not be described here.
Just as those implementations, it computed the Gram-Schmidt factors in floating
point approximations, and not in exact rational arithmetic, to make
the computations practical.
For each initial basis, several iterations were performed; after reducing
a given basis, the
rows of the reduced basis were permuted, and the Lov\'{a}sz algorithm
was applied to that basis.
This had roughly the same effect as the procedure followed in \cite{LO2}, in which several
permutations of the initial basis were reduced
separately, in that additional reductions gave sometimes better and sometimes worse results.

As in \cite{LO2,OtR}, the lattice basis reduction algorithm was implemented
using Brent's MP multiple precision package \cite{Br4}.
The lattice basis of the form \eqn{eq323} to which it was applied usually had $40 \le n \le 85$, $70 \le m \le 75$, and $11 \le  r \le 16$,
and usually about 6 successive reductions were performed.
All the values of $t$ from all the reductions (several thousand values in total) were collected and used to compute $|P_Y (t)| $ with $Y$
about
$p_{95} = 499$.
Those $t$ for which $|P_Y (t)| $ was largest (in a given range of values of $t$)
were then used for the computations described in Section~\ref{sc31}.
\section{Possible extensions}\label{sc33}
One possible way to obtain even better values of $t$
is to speed up the implementation of the Lov\'{a}sz
algorithm.
The Brent MP package \cite{Br4} was written to be portable and is not
very efficient, and on a machine like the Cray~X-MP is about 10 times slower
than a program customized for this machine could be.
Also, there are some nice methods for speeding up the
Lov\'{a}sz algorithm
itself that have been developed by Radziszowski and Kreher \cite{RK}.
All these improvements could be used to reduce lattices of larger dimensions or reduce more permutations of a given basis.
Another approach might be to develop better lattice basis reduction algorithms.
Several approaches are available, such as those of Schnorr \cite{Sch1,Sch2}, but
apparently none of them have been implemented yet.
Any one of those approaches could also be combined with simpler tricks, such as
that of trying to maximize a product like that of \eqn{eq322}, but where some of the large primes are replaced by slightly larger primes.

All the above approaches have major limitations.
Logarithms of primes are rationally independent, and ought to behave
like independent random variables
as far as
multidimensional
diophantine approximation
properties
are concerned.
This means that given any fixed subset $S$ of them, values of $t$ for which
all the $t \log p$ 
for $p \in S$ 
are small modulo$~2 \pi$
are likely to be far apart,
and if $S$ is large, the smallest value of $t$ of this kind is likely to be large.
Therefore to find values of $t$ for which $\zeta (1/2 +it)$ is large, we probably need algorithms that can find vectors in extremely high dimensional
lattices that are only slightly shorter than usual, as
opposed to the method that has been used, which finds very short
vectors in low dimensions.
It is doubtful that any of the approaches suggested above could yield such algorithms.

Computations with some of the values of $t$ that were found during the main computations of Section~\ref{sc2} and for which $\zeta (1/2 +it )$ is large confirm the suggestion
above that such large values arise typically from unpredictable interactions of many large primes and not from an almost perfect lining up of a small set of initial primes.
Therefore further searches for values of $t$ with $\zeta (1/2 + it )$ large using
algorithms known or foreseeable today might produce additional interesting
phenomena, but is not likely to find all the large values.

Simultaneous
diophantine approximation algorithms
could also be applied to find other values of $t$ for which $\zeta (1/2+ it )$ is unusual.
For example, the values of $t$ in Table~3.2.1 all lie close to values of $u$ for which
$S(u)$ is large, but that is a by-product of having a large gap
between zeros in that region.
One could also try to search directly for values of $t$ for which $S(t)$ is large.
There are various formulas for $S(t)$, such as those of Selberg (Theorem~14.21 of \cite{Tit2})
or Goldston \cite{Go2} (see Section~\ref{sc26}).
The main term in Selberg's formula suggests that to make $S(t)$ large, we
ought to search for $t$ such that
\beql{eq331}
\sum_{p \le X}~ \mbox{Arg} ( 1 -  p^{- 1/2 -it} )
\eeq
is large in absolute value.
This task can be formulated easily as a diophantine approximation problem,
but to obtain large values,
it appears that
we need to deal with large $X$,
which tends to produce impracticably large values of $t$.
There are two culprits here.
One is that the contribution of the sum in \eqn{eq331} to $S(t)$ is divided by $\pi$.
The other one is that the error term in Selberg's formula is large compared to the main term in ranges of $t$ that are of interest.
(This is to be expected, since $|S(t)| < 2.9$ for all values that have been computed,
while the remainder terms in Selberg's formula have to produce the jumps by 1 of $S(t)$ at zeros,
since the main term is continuous.)

The final conclusion to be drawn from the above discussion is that searches for special values of $\zeta (1/2 +it)$ do produce interesting results and can be improved
somewhat, but there is no method in sight that is likely to produce all the points of interest.

%  FILE: ch4.tex

\chapter{Algorithms and their implementation}\label{sc4}
\section{Introduction}\label{sc40}
\hspace*{\parindent}
The main result of \cite{OS}, namely Theorem~5.1,
can be reformulated for the case of computations of $\zeta (1/2 +it)$ as follows:
\begin{quote}
{\em
For any $a \in [0, 1/2]$ and any positive constants $\delta$ and $c_1$, there is an
effectively computable constant $c_2 = c_2 ( \delta , c_1 , a)$ and an algorithm that for every $T > 0$ will
perform $ \le c_2 T^{1/2 + \delta}$ operations on numbers of
$\le c_2 \log T$ bits using $ \le c_2 T^{a+ \delta}$ bits of storage
and will then be capable of computing any value $Z(t)$ for $T \le t \le T + T^a$ to within $\pm T^{-c_1}$ in $\le c_2 T^\delta$ operations
using the precomputed and stored values.
}
\end{quote}
This result is completely rigorous, but implementing it as it is described in \cite{OS} presents difficulties because of the need for high precision and large storage.
This section shows a modified version of the algorithm that is practical, but which does sacrifice some of the rigor of the basic result to achieve speed.
Many of the choices that were made in the implementation were forced or at least
suggested by the hardware and software that was used, and would have been made differently on another machine.

All the main computations were carried out on a Cray X-MP supercomputer with 2 processors and 4 million words of main memory.
Although occasionally both processors were used, there was no true parallel processing involved, as the programs did not interact with each other.
The Cray computers have 64-bit words, with 48-bit mantissas
(including the sign bit), which give slightly over 14 decimal digits of precision in the standard single precision $(sp)$
floating point numbers, and slightly over 28 decimal digits in double precision $(dp)$.
(See \cite{Od2} for a more extended discussion of this issue.)
A crucial part
of the algorithm, as will be shown later, involves computing
$\exp (it \log n)$for $n$ ranging up to about $t^{1/2}$.
Since $t \approx 1.5 \times 10^{19}$
near the $10^{20}$-th zero,
$t \log n$ is on the order of $10^{20}$,
and so if we do the computations in $dp$, then after reducing modulo $2 \pi$ we are left with only about 8 decimal digits of accuracy,
and this is also true for values of $\exp (it \log n)$ that we obtain
after exponentiating.
This is only barely acceptable, and accounts for most of the lack of rigor in the computations.
Attempting this computation in $sp$ would produce a totally meaningless answer.
On the other hand, the Cray is designed for $sp$ computations that vectorize.
All $dp$ computations are done in software, and
although some of them are vectorized by the latest Cray compilers,
some
$dp$ arithmetic operations
are about
100 times
slower than $sp$ ones.
Therefore even though $dp$ computations were by themselves only barely accurate enough, it was necessary to do as much computing as possible in $sp$ to obtain
high speed.
To achieve this, some hybrid methods described in Sections~\ref{sc41} and \ref{sc42} were used.

The problems outlined above of getting sufficient accuracy were due not to the nature
of the new algorithm but to the large height at which the computations were
undertaken.
Implementation of any of the older algorithms (such as that of the Riemann-Siegel formula discussed in Section~\ref{sc41}) would have had to cope with the same difficulties.
(No matter which algorithm was used, supercomputers like the Cray would be essential in practice, since less powerful machines typically have only 32-bit words, which would
require using multiple precision packages, which are prohibitively slow.)
The new algorithm does introduce some additional sources of errors, however, which would make rigorous error analysis harder than it would be for the older methods even if higher precision computations were employed.

The present implementation applies only to computations of the zeta function on the critical
line.
The algorithm of \cite{OS} can also be used to compute the zeta function on other lines,
and this has applications to problems such as that of computing $\pi (x)$ \cite{LO4},
but no attempt was made to write programs to carry out such applications.
The method of \cite{OS} also applies to the computation of Dirichlet $L$-function
and related functions.
Only minor modifications to the present implementation would be
needed to compute Dirichlet $L$-functions, and this may be done in the future.

The main computations were all carried out on a Cray X-MP supercomputer running the
Unicos~2.0 operating system,
with some of the final statistical computations done under Unicos~3.0.
The language of the main computations was Fortran, with
several
different compilers being used.
Various UNIX\tm~ tools, such as the Awk programming language \cite{AKW},
were utilized.
Many of the statistical studies of zeros were carried out on a DEC~VAX 8550 computer using Fortran, Awk, or (especially) the S statistical programming language \cite{BC}.
S was also used to produce all the graphs in this paper.
\section{Zero-locating program}\label{sc41}
\hspace*{\parindent}
The program for locating zeros is based on the Riemann-Siegel formula
\cite{Ed,Gab,Iv,Sie1,Tit2},
which has been the basic tool for all zeta function computations at large
height during the last 60 years.
This formula says that if
\beql{eq411}
\tau = t/(2 \pi ) ,~ k_1 = \left \lf \tau^{1/2} \right \rf , ~
z = 2 ( \tau^{1/2} - k_1 ) - 1 ~,
\eeq
then for any $m \ge 0$,
\beql{eq412}
\begin{array}{l}
Z(t) = 2  \ds_{k=1}^{k_1}  k^{-1/2} \cos ( t \log k - \theta (t)) \\
~~ \\
~~~~~~~~~~~~~ +~ (-1)^{k_1 +1} \tau^{-1/4} \ds_{j=0}^m
\Phi_j (z) (-1)^j \tau^{-j/2} + R_m ( \tau ) ~,
\end{array}
\eeq
where the $\Phi_j (z)$ are certain entire functions
that can be expressed in terms of derivatives of
$$
\Phi_0 (z) = \frac{\cos \{ \pi ( 4z^2 + 3)/8 \} }{\cos ( \pi z )} ~,
$$
and
\beql{eq413}
R_m ( \tau ) = O( \tau^{-(2m+3)/4} ) ~~~\mbox{as} ~~~
\tau \to \In ~.
\eeq

Gabcke \cite{Gab} has obtained essentially optimal bounds for the remainder terms $R_m ( \tau )$, and the one used in the new computations was
\beql{eq414}
| R_1 ( \tau ) | \le 0.053 t^{-5/4} ~~~\mbox{for} ~~~
t \ge 200 ~.
\eeq
The asymptotic expansion terms $\Phi_0 (z)$ and $\Phi_1 (z)$ were computed using their
Taylor series expansions \cite{CR,Gab}.

The main difficulty in computing $Z(t)$ using the Riemann-Siegel
formula is in the evaluation of the cosine sum in \eqn{eq412}.
(For $t$ near the $10^{29}$-th zero, $k_1 \approx 1.5 \times 10^9$.)
In the new implementation it was computed as the sum of two terms,
\beql{eq415}
Z_1 (t) = 2  \sum_{k=1}^{k_0 -1} ~ k^{-1/2} \cos ( t \log k - \theta (t)) ~,
\eeq
and
\beql{eq416}
\mbox{Re} ~e^{-i \theta (t)} F(t) ~,
\eeq
where
\beql{eq417}
F(t) = F(k_0 , k_1 ; t) = \sum_{k=k_0}^{k_1}
2k^{-1/2} \exp ( it \log k ) ~.
\eeq
The advantage of the new algorithm over the straightforward term-by-term
evaluation of the Riemann-Siegel formula is in the method of evaluating $F(t)$,
which is an adaptation of the method presented in \cite{OS}, and is described in detail in Sections~\ref{sc42} and \ref{sc43}.
We will now describe the computations of $\theta (t)$, of $Z_1 (t)$,
and of the zero-locating procedure.

One could take $k_0 =1$, in which case $Z_1 (t) =0$ identically,
but for technical reasons having to do with the speed of
the algorithm for computing $F(t)$ it was advantageous not to do this, and in practice one had $100 \le k_0 \le 500$.
(See Table~4.5.1 for some values.)
The method used to compute $Z_1 (t)$ was essentially the same as that used in \cite{Od2}
for computing the entire cosine sum in the Riemann-Siegel formula.
The argument $t$ was always maintained as a $dp$ variable.
Another $dp$ variable, $t_0$,
was also maintained, which normally had the property that $| t- t_0 | \le 10$.
Three arrays, $d_n , q_n , u_n$, $1 \le n \le k_0 -1$, were also used;
$d_n$ was the $dp$ value of $\log n$, $q_n$ was the value of $2n^{-1/2}$, computed in $dp$ but stored in $sp$,
and $u_n$ was the value of $t_0 \log n$ reduced modulo $2 \pi$, where the computation was again done in $dp$ but the stored value
was in $sp$.
To compute $Z_1 (t)$ for a new value of $t$,
$t$ was compared to $t_0$.
If $|t - t_0 | > 10$,
$t_0$ was set to $t$, and the $u_n$ were recomputed.
At that point (and also if $|t- t_0 | \le 10$ was satisfied initially)
$\delta$ was defined as the $sp$ value of $t-t_0$, $t_1$ as the $dp$
value of $t_0 + \delta$, $\theta (t_1 )$ was computed in $dp$,
reduced $\bmod~2 \pi$, and converted to an $sp$ variable $v$.
Finally,
$Z_1 (t)$ was computed as the sum (in $sp$) of
$$w_n = q_n \cos ( \delta e_n + u_n -v ) , ~~~
1 \le n \le k_0 -1 ~,$$
where $e_n$ is the $sp$ value of $d_n$ (obtained by truncation).

For the computation of $\theta (t)$, another $dp$ variable $\tilde{t}_0$ was maintained together with the $dp$ value of $\theta ( \tilde{t}_0 )$ and with $dp$ or $sp$ (depending on order) values of
derivatives of $\theta ( t)$ at $\tilde{t}_0$.
When $|t- \tilde{t}_0 | \le 50$ was satisfied, $\theta (t)$ was computed
from the stored values using its Taylor series expansion around
$\tilde{t}_0$, using partially $dp$ and partially $sp$ arithmetic.
When $| t- \tilde{t}_0 | > 50$,
$\tilde{t}_0$ was set to $t$ and $\theta (t)$ and its derivatives were computed
in $dp$ (or $sp$ for higher derivatives) using Stirling's formula.
The reason for this involved procedure was to avoid using the Cray
$dp$ logarithm function, which was extremely slow when the program was
being written. Later, a new version of the $dp$ logarithm
routine was installed in the system libraries that is about $4$ times
faster than the old one,
so that this procedure does not gain much.
However, the old procedure
was retained, both because it was still faster, and
because of the considerations of accuracy and reliability
of the computational results that are described in Section~\ref{sc45}.

The procedure for locating zeros was the standard one of finding Gram blocks
and searching for the expected number of sign changes of $Z(t)$ in them.
When a violation of Rosser's rule was encountered,
the program searched neighboring Gram blocks.
Once all the zeros were separated, they were located to a nominal accuracy
(i.e, disregarding any inaccuracy in the computation) of $\pm 2 \times 10^{-8}$ by the Brent combination \cite{Br1} of linear and quadratic interpolation.
The sophisticated zero-locating strategies of \cite{LRW1,LRW2} were not employed,
and about 8.5 evaluations of $Z(t)$ were used on average to compute each zero.
(An additional 1 evaluation of $Z(t)$ per zero was performed to determine the value of $Z(t)$ halfway between zeros.)
\section{Odlyzko-Sch\"{o}nhage algorithm}\label{sc42}
\hspace*{\parindent}
The function $F(t) = F(k_0 , k_1 ; t)$ is computed in two stages.
In the first, precomputation stage, which accounts for most of the computing time,
$F(t)$ is computed at a uniform grid of points.
\beql{eq421}
t= T, ~T + \delta , \dd ,~T+ (R-1) \delta ~.
\eeq
The second stage, described in Section~\ref{sc43}, is fast, and computes the values of $F(t)$ for
$T+A \le t \le T+(R-1) \delta -A$ for a certain constant $A$ from the
stored values of $F(T)$, $F(T+ \delta ) , \dd , F(T+(R-1) \delta )$.
This section describes the precomputation phase.
It is based on \cite{OS} with only minor modifications,
and although it is essentially complete, it is technical.
The description in \cite{OS}
does not cover the details of the implementation, but is more conceptual and
easier to read, and is therefore likely to be preferable
for those interested only in the basic
ideas of the algorithm and not in the details.

Let $r \in Z^+$, and define
\beql{eq422}
R= 2^r , ~~\omega = \exp ( 2 \pi i / R ) ~.
\eeq
In principle any $R$ for which the Fast Fourier Transform ({\em FFT}) can be applied efficiently could be used, but it was
convenient to work with powers of 2.
The values of $r$ that were used in the main computations were $r = 17, 19, 23$, and 24.

For $-R/2 \le h < R/2$, define
\beql{eq423}
u_h = \sum_{j=0}^{R-1}  F(T+ j \delta ) \omega^{-hj} ~.
\eeq
Once the $u_h$ are computed, the $F(T+ j \delta )$ can be obtained from them fast through the FFT:
\beql{eq424}
F(T+ j \delta ) = R^{-1}  \sum_{h=-R/2}^{R/2-1} u_h \omega^{jh} ~.
\eeq
This computation takes a negligible amount of time.

Using the definition \eqn{eq417} of $F(t)$ in \eqn{eq423}, exchanging the orders of summation,
and summing the geometric series that arises, one obtains
\beql{eq425}
u_h = \omega^h  \sum_{k=k_0}^{k_1} \frac{a_k}{\omega^h - b_k} ~,
\eeq
where the $\beta_k$ are defined so that $-R/2 \le \beta_k < R/2$ and
\begin{eqnarray}
\label{eq426}
b_k & = & \exp ( 2 \pi i \beta_k /R ) = \exp ( i \delta \log k ) ~,
\\
\label{eq427}
a_k & = & 2 k^{-1/2} e^{iT \log k} (1-e^{iR \delta \log k} ) ~.
\end{eqnarray}
Write
\beql{eq428}
f(z) = \sum_{k=k_0}^{k_1} \frac{a_k}{z- b_k} ~,
\eeq
Then we need to evaluate $f( \omega^h )$ for $-R/2 \le h < R/2$.
Term-by-term evaluation of the sum in (4.2.8) would require on the
order of $k_1 R$ operations, which of the same complexity
as evaluating the Riemann-Siegel formula in the standard way at each point $T+j \delta$.
However, the new algorithm
of \cite{OS} leads to much faster evaluation of the $f( \omega^h )$
by means of Taylor series expansions.
Let $\langle x \rangle$ denote the nearest integer to $x$,
let $\| x \|_{R}$ denote the ``cyclic distance''
modulo $R$:
$$\| x \|_{R}
= \min_m  | x - mR | ~,$$
and for integers $p, q$ with $q \ge 0$,
$3^q \le R/2 +1$, $-R/2 \le p < R/2$,
$| p 3^q | \le R/2 -1 + (3^q -1 ) /2$, define
\beql{eq429}
I_{p,q} =
\left \{ k : k_0 \le k \le k_1 , 
\| \beta_k - p 3^q \|_{R} \ge 3^q -1 ,
\| \beta_k - \langle p/3 \rangle 3^{q+1} \|_{ R} < 3^{q+1} -1 \right \} \,.
\eeq
Then it can be shown easily ({\em cf.}~\cite{OS}) that each
$k$ belongs to at most 6 different $I_{p,q}$ for a fixed $q$.

Let $Q = \left \lf \log_3 ( R/2 +1) \right \rf$.
Then for any $h$,
$-R/2 \le h < R /2$, it is easy to see
({\em cf.}~\cite{OS}) that
$\{ k_0 , k_0 +1 , \dd , k_1 \} $ is the disjoint
union of the sets $I_{\langle h3^{-q} \rangle , q}$ for
$0 \le q \le Q$.
Hence if
\beql{eq4210}
f_{p,q} (z) = \sum_{k \in I_{p,q}} \frac{a_k}{z - b_k}~,
\eeq
then for $-R/2 \le h < R/2$, we have
\beql{eq4211}
f ( \omega^h ) = \sum_{q=0}^Q  f_{\langle h3^{-q} \rangle , q} ( \omega^h ) ~.
\eeq
The new algorithm evaluates the functions $f_{p,q} (z)$ at points
$z= \omega^h$ with $\langle h3^{-q} \rangle = p$.
For $q \le Q_1$, ordinary evaluation of the sum in \eqn{eq4210} is used.
For $Q_1 < q \le Q$, the function $f_{p,q} (z)$ is expanded in its Taylor series around the point
\beql{eq4212}
z_{p,q} = \exp ( 2 \pi i p 3^q /R ) ~.
\eeq
It is easy to show
({\em cf.}~\cite{OS}) that these Taylor series converge fast, so not too many
terms in them have to be kept.
Finally, these Taylor series are used to evaluate the $f_{p,q} ( \omega^h )$.

The threshold $Q_1$ was taken to be 3 in all the computations after some experiments showed that it was reasonably close to the optimal choice.
The Taylor series method is inefficient when
$|I_{p,q} | $ is small, since its overhead is large.
A slight improvement in the program could be obtained by selecting which
method to use based on $| I_{p,q} | $ and not on $q$ alone.

The main computation proceeds in stages indexed by integers $m$,
$-R/2 \le m \le R/2-1$.
In stage $m$, only $k \in S_m$ are considered, where
\beql{eq4213}
S_m = \{ k :
k_0 \le k \le k_1 ,~ \beta_k \in [m, m+1 ) \}~ .
\eeq
For any $p$ and $q$, if $k \in I_{p,q}$ for some $k \in S_m$,
then $S_m \subseteq I_{p,q}$, which makes bookkeeping for the various computations easy.
The distribution of the $\beta_k$ is nonuniform, with almost all the time being spent in the small fraction of stages $m$ for which $|S_m | $ is large.

Each stage $m$ is further subdivided into substages corresponding to a partition of $S_m$ into blocks $S_{m,j}$,
$1 \le j \le s$, of consecutive $k$'s with $|S_{m,j} | \le 2560$ for all $j$,
and $|S_{m,j} | < 2560$ being possible only for $j=s$.
(For almost all stages $|S_m | \le 2560$, and so $s=1$.)
This was done to keep the sizes of the auxiliary arrays small, and also to have their lengths be
multiples of 64, the length of Cray vector registers.

Suppose that
$$S_{m,j} = \{ k :~ k_2 \le k \le k_3 \}~ .
$$
Several auxiliary arrays are defined.
The most important and most time consuming to compute
is the $d_k$ array, $k_2 \le k \le k_3$,
with $d_k$ being an approximation to the $dp$ value of $\log k$.
The set $S_{m,i}$ is partitioned into blocks of 64 consecutive values of
$k$ (with the last block possibly being smaller), and if a block consists of $k$'s with $k_4 \le k \le k_5 \le k_4 + 63$,
then $d_{k_5}$ is computed
using the Cray $dp$ logarithm routine, and the
$d_k$, $k_4 \le k < k_5$ are then computed from $d_{k_5}$ using
Taylor series expansions.
When the program was first written, this involved procedure was about 6 times faster (for computations
near zero number $10^{20}$) than using the Cray $dp$
logarithm routine, which served to cut the running time of the entire rational evaluation program
by over 30\%.
(As a result, the computation of the $d_k$ now takes
about
10\% of the total
running time instead of the roughly half that was required by the earliest version
of the program which involved the Cray $dp$ logarithm function.)
The latest Cray mathematical subroutine libraries have a $dp$ logarithm routine
that is about 4 times faster than the old one,
and so the procedure described above is only about 1.5 times
as fast as using the standard Cray $dp$ logarithm all the time would be.
(Much faster variants of this method are possible, as is shown in Section~\ref{sc461}.)

Once the $d_k$ are computed, they are used to calculate $T \log k$
$ \mbox{modulo}~2 \pi $ in $dp$, which is then
converted to $sp$ and used to compute $\exp (iT \log k)$ utilizing the Cray
cosine and sine routines.
The $2k^{-1/2}$ factor is also computed in $sp$ arithmetic.
Finally the difference $1 - \exp (iR \delta \log k )$ is computed in the form
\beql{eq4214}
-2i \exp \lt i \frac{1}{2}  R \delta \log k \rt  \sin \lt \frac{1}{2}  R \delta \log k \rt ~,
\eeq
where $sp$ arithmetic is used for the trigonometric functions,
but $2^{-1} R \delta \log k$ is computed in $dp$ and reduced modulo $2 \pi$
in $dp$, for reasons that will be explained later.
All these factors are then combined using $sp$
arithmetic to obtain $a_k$.
The $b_k$ are also computed in $sp$.

For $q=2$ and 3, ordinary complex $sp$ arithmetic is used to evaluate
the $a_k / ( \omega^h - b_k )$ for $k \in S_{m,j}$ 
and these are added to stored variables corresponding to $f( \omega^h )$.
For $q \ge 4$, complex $sp$ arithmetic is used to compute the coefficients
$a_k ( z_{p,q} - b_k )^{-n-1}$ for $0 \le n \le V$
in the Taylor series expansion
\beql{eq4215}
\frac{a_k}{z - b_k} = \sum_{n=0}^\In  a_k ( z_{p,q} - b_k )^{-n-1} ( z_{p,q} - z )^n
\eeq
around $z_{p,q}$, and these are added to the arrays holding the Taylor series coefficients of $f_{p,q} (z)$.
The number of terms $V$ depends on $m, p, q$, and is chosen
so as to make the $V$-th computed coefficient about $10^{-15}$ times the size of the $0$-th
one.
Except for $q$ close to $Q$, $V$ is almost always
$< 50$.
After all the $S_m$ have been processed, the Taylor series of the
$f_{p,q} (z)$ are used to compute the $f_{\langle h3^{-q} , q \rangle} ( \omega^h )$ in $sp$ arithmetic for $q \ge 4$, and these numbers are then added to the
variables corresponding to $f( \omega^h )$.
Since the $a_k$ are only accurate to 9 or 10 decimal digits in
the computations near zero number $10^{20}$, one could
take $V$ much smaller for computations at such large heights,
say about
2/3 of the present value, without
significantly
affecting the accuracy of the final results.
This would speed up the main program by about 15\%.
This modification was not made in the programs to keep them the same for all heights.

For $q=0$ and 1, a special procedure is used, since here $b_k$ and $z_{p,q} = \omega^h$ (for $h=p3^q$) are close to each other,
and so computing $z_{p,q} - b_k$ in $sp$ would lead to
large errors.
Instead, we use the expansion
\beql{eq4216}
\omega^h - b_k = -2i \exp ( \pi i (h+ \beta_k )/R ) \sin ( \pi (h- \beta_k ) /R ) .
\eeq
The $\pi ( h- \beta_k )/R$ factor is evaluated in $dp$, reduced modulo $2 \pi$, and converted to $sp$ before being used to evaluate the sine.
If $(h- \beta_k ) /R$ is small,
the definition \eqn{eq426} of $\beta_k$ shows that $2^{-1} R \delta \log k$ reduced modulo
$2 \pi$ cannot be too large, and the ratio of the two sines in \eqn{eq4214} and \eqn{eq4216}
is bounded by $R$ in absolute values.
The Cray $\sin (x)$ routine is accurate for small $x$, since it 
computes
$x( \sin (x) /x )$, and so the quotient of the quantities in \eqn{eq4214}
and \eqn{eq4216} is evaluated accurately.
(The computation of $2^{-1} R \delta \log k$ modulo $2 \pi$, which was mentioned
above, is done in $dp$ to make sure that the arguments of sine in these
computations are accurate.)

Aside from the $dp$ operations, which are often not vectorized by the Cray compilers, most of the computations
were written so they would be vectorized automatically by the compiler.
(No assembly language routines were used.)
This is even true of the Taylor series expansions, since those are almost always
performed on large sets of $k$'s
simultaneously, so the inner loops are written to run on $k$, and not on the index of the Taylor
series term being evaluated.
(This does require the use of some auxiliary arrays, but since at most 2560 $k$'s
are considered in each stage, storage in not a problem.)
As will be described in Section~\ref{sc461}, some of the crucial loops in the
program are executed at the rate of over 100 million floating
point operations per second,
which is fast for Fortran programs, since the cycle time on the Cray~X-MP is 9.5 nanoseconds.

The above sketch of the implementation of the rational function evaluation
algorithm applies directly only for the runs with $r=17$ and 19.
For $r=23$ and 24, a modified version of the algorithm had to be used
because of space restrictions that are discussed at greater
length in Section~\ref{sc44}.
In the implementation discussed above, a complex array of length $R$ is kept for the values
$f( \omega^h )$,
$-R/2 \le h < R/2$, as well as arrays for the Taylor series coefficients
of the $f_{p,q} (z)$ with $q > Q_1 = 3$.
In the versions used for $r \ge 23$, the program works on $2^{r-17}$ segments of values of $h$, each of length $2^{17} = 131072$.
If we denote one such segment by
$$H= \{ h :~ h_0 \le h < h_0 + 2^17 \} $$
$(h_0 = -R/2$, $-R/2 + 2^{17}$, etc.), then the main program computes
the contribution to $f( \omega^h )$ for $h \in H$ of all $k$ such that $k \in I_{p,q}$ with some $0 \le q \le 8$,
$p = \langle h' 3^{-q} \rangle$ for some $h' \in H$,
where these contributions are computed as before,
namely directly for $q \le Q_1$, and through Taylor series expansions for
$q > Q_1$.
These values are stored in a file.
Another file is also created, which contains the contributions to the Taylor
series coefficients for $q \ge 9$ of all the
$$k \in \bigcup_{m \in H}  S_m ~.
$$
As different $H$'s are processed, the Taylor series contributions for $q \ge 9$ are
added, and at the end they are combined with the previously computed
contributions of $q \le 8$ to obtain the values of $f( \omega^h )$.

The algorithm is involved, and its running time depends on a complex combination of various factors.
A rough indication of where most of the time is spent is provided by
Table~4.3.1.
It is based on experiments with the algorithm for $r=17$,
when it is applied to evaluate the $f( \omega^h )$ for
$k_0 \approx 1.5 \times 10^9$, $k_1 = k_0 + 10^6$,
$T \approx 1.5 \times 10^{19}$, $\delta = 0.15$.
The total running time
was 132 seconds.
The figures in Table~4.3.1 should be treated with caution as only a rough
indication of where most of the computational effort was spent.

The basic FFT routines that were used were those of Bailey \cite{Bail}.
They were written especially for the Cray-2, where they are both faster and more
accurate than the standard Cray routines.
On the Cray~X-MP, Bailey's routines are slightly slower than the standard Cray ones.
They were selected because of their greater accuracy, although in
comparison with
the errors in the rational function evaluation algorithm, the additional errors introduced by the FFT program are negligible.
The time needed for the FFT itself was completely negligible, with
complex transform on $2^{19}$ points taking under 1 second, much less time
than it took to read in the data.

Because of
space limitations on the Cray~X-MP, Bailey's routines could be used directly
only for $r=17$ and 19.
For $r=23$ and 24 it was necessary to perform extensive reformatting operations.
Suppose that we wish to take the FFT of $v_0 , \dd , v_{M-1}$, where $M = 2^g K$, say,
and we can only carry out FFT of length $K$ in core.
If $w_0 , \dd , w_{M-1}$ is the Fourier transform of the $v_j$, then
\begin{eqnarray}
\label{eq4217}
w_h & = & \sum_{j=0}^{M-1}  v_j \exp ( 2 \pi i hj /M ) \nonumber \\
& = & \sum_{s=0}^{2^g -1} \exp ( 2 \pi i h s/M) 
\sum_{m=0}^{K-1} v_{2^g m+s} \exp ( 2 \pi i mh/K ) ~.
\end{eqnarray}
The inner sum above is just the Fourier transform of a sequence of length $K$, and can be handled by the FFT directly.
To implement this, one needs to create new data sets consisting of the subsequences $v_{2^g m+s}$,
$0 \le m \le K-1$, carry out the FFT on them, and then combine them to obtain the $w_h$.
For
computations with $r=23$, for example,
Bailey's algorithm is used with $K= 2^{19}$, so that for each of the
$16=2^4$ FFT's, all $2^{23}$ values $v_j$ have to be read, and after all the FFT's are done, 16 passes through the data are performed to compute
and store the decimal linear combination given by \eqn{eq4217}.
This takes
about
an hour of elapsed time (the exact length depending on the load on the system), although very little computing time.
For $r=24$, the total time is about 4 times longer.
For large computations, it would be worthwhile to use
more efficient procedures, some of which are discussed in Section~\ref{sc46}.
Such procedures would have been advantageous even for computations on the scale described here, and the only reason they were not carried out was
the
additional programming effort
that would have been required, and the limited facilities for data storage
that were available.
\section{Band-limited function interpolation}\label{sc43}
\hspace*{\parindent}
Section~\ref{sc42} shows how $F(t)$,
$F(T+ \delta ) , \dd$, $F(T+(R-1) \delta )$ are computed.
In general, though, we need to compute $F(t)$ for various $t \in (T, T + (R-1) \delta )$ that are not predictable a priori.
The approach that was presented in \cite{OS} was to compute several of the derivatives
$F^{(h)} (t)$ at the grid points
$t=T$, $T+ \delta , \dd$, $T + (R-1) \delta$,
and then to compute desired values
of $F(t)$ by expanding in a Taylor series around the nearest grid point.
Since the derivatives $F^{(h)} (t)$ are representable as sums similar to that for $F(t)$,
they can be computed by a variant of the algorithm described in Section~\ref{sc42}.
However, the need to use a dense grid
and to store the derivatives $F^{(h)} (t)$ at the grid points make this approach
inefficient.
Another possible approach is that of
interpolating values of $Z(t)$ from the values computed on the grid
$T, T+ \delta , \dd , T + (R-1) \delta$, as is done in \cite{Hej5}, for example,
where $Z(t)$ is approximated as if it were a polynomial through the Lagrange interpolation formula.
This method also appears inefficient, and furthermore it is not
rigorous.

The method that is used to compute $F(t)$ for $t$ not a grid point is based on band-limited function interpolation techniques.
If
\beql{eq431}
G(t) = \int_{- \tau}^\tau  g(x) e^{ixt} dx ~,
\eeq
then it's been known for a long time that $G(t)$ is determined by its samples at the points
$n \pi / \tau$, $n \in Z$, provided only that $G(t)$ satisfies some mild conditions,
and further that $G(t)$ is then representable by the ``cardinal series''
\beql{eq432}
G(t) = \sum_{n=- \In}^\In  G \lt \frac{n \pi}{\tau} \rt 
\frac{\sin ( \tau t - n \pi )}{\tau t - n \pi} ~.
\eeq
Results of this type have a long history, going back to E. Borel, Hadamard,
de~la~Vall\'{e}e Poussin,
E.~T. and J.~M. Whittaker, and
Ferrar in the mathematical literature, and to Nyquist, Kotelnikov, Shannon, and Someya in engineering (see \cite{Hig} for a history), and are the basis for digital sound transmission and storage, for example.
Two comprehensive surveys of the literature in this area are those of
Butzer et~al. \cite{BSS} and Jerri \cite{Jer}.

The cardinal series in \eqn{eq432} is not suitable for the interpolation of $F(t)$ because, aside from the question of whether the expansion \eqn{eq432} is valid for $F(t)$, the sum in \eqn{eq432} converges slowly.
We use instead a formula for $G(t)$ that involves a sum of $G(n \pi / \beta )$ for some $\beta > \tau$,
which thus involves more frequent (and thus less efficient) sampling of $G(t)$,
but in which the coefficients of $G( n \pi / \beta )$ decrease rapidly.
The basic approach appears to be well-known to many analysts and communications engineers, but no published reference for the result we use was found, so a proof is
sketched below.
(See \cite{BSS,Jer} for other possible approaches.)

Suppose that $G(t)$ satisfies \eqn{eq431}, where $g(x)$ will be assumed for the moment to be
in $L^2 (- \tau , \tau )$.
Take $\beta > \tau$ and define $g(x) =0$ for $\tau < | x | < \beta$, and then extend $g(x)$ to the entire real line by making it periodic with period $2 \beta$.
Then we have
\beql{eq433}
g(x) = \sum_n  a_n  \exp ( 2 \pi in x / ( 2 \beta )) ~,
\eeq
where
\beql{eq434}
a_n = ( 2 \beta )^{-1}  \int_{- \beta}^\beta  g(x) \exp
(-2 \pi in x / ( 2 \beta )) dx ~.
\eeq
Eq.~\eqn{eq431} then shows that
\beql{eq435}
a_n = ( 2 \beta )^{-1}  G( - n \pi / \beta ) ~.
\eeq
Next, choose $\lambda , \epsilon > 0$ so that
\beql{eq436}
\tau \le \lambda - \epsilon < \lambda + \epsilon \le \beta ~,
\eeq
and let $H(x)$ be some continuous function with $H(x) =0$ for $|x| > \epsilon$, and
\beql{eq437}
\int_{- \In}^\In  H(x) dx = 1 ~.
\eeq
Further, let $\chi (x)$ be the characteristic function of the interval $[- \lambda , \lambda ]$, and let $u star v$ denote the convolution of
the functions $u$ and $v$:
$$(u \ast v) (x) = \int_{- \In}^\In  u(y)
v(x-y) dy ~.$$
Then
\beql{eq438}
( \chi \ast H) (x)
= \int_{x- \lambda}^{x+ \lambda} H(y) dy = \left\{
\begin{array}{ll}
1, & |x| \le \lambda - \epsilon~, \\
~~ \\ [-.09in]
0, & |x| \ge \lambda + \epsilon~.
\end{array}
\right.
\eeq
Therefore
\beql{eq439}
G(t) = \int_{- \tau}^\tau  g(x) e^{ixt} dx = \int_{- \In}^\In 
g(x) e^{ixt} ( \chi \ast H) (x) dx ~.
\eeq
Substituting the Fourier series
\eqn{eq433}
into the last expression above and using \eqn{eq435}
yields
\beql{eq4310}
G(t) =
(2 \beta )^{-1} \sum_n  G( -n \pi / \beta ) \int_{- \In}^\In 
e^{ixn \pi / \beta + ixt} ( \chi \ast H) (x) dx ~.
\eeq
If we change $n$ to $-n$ in the above formula, then the
integral above is just the Fourier transform of $\chi \ast H$ evaluated at $n \pi / \beta - t$,
which is the product of the Fourier transforms of $\chi$ and $H$.
If $h(t)$ is the Fourier transform of $H(x)$,
\beql{eq4311}
h(t) = \int_{- \In}^\In  H(x) e^{ixt} dx ~,
\eeq
then we obtain
\beql{eq4312}
G(t) = \frac\lambda\beta  \sum_n  G( n \pi / \beta ) 
\frac{\sin \lambda ( n \pi / \beta -t)}
{\lambda (n \pi / \beta -t )} ~ h( n \pi / \beta - t ) ~.
\eeq
The interpolation formula \eqn{eq4312} was derived under the assumption that
$g(x) \in L^2 (- \tau , \tau )$, but by taking limits,
it is easy to see that this formula holds when $g(x)$ is a finite linear combination of
delta functions, as well as in more general settings.

The formula \eqn{eq4312} can be applied directly with $G(t) = F(t)$ for $\tau = \log k_1$,
but since the spectrum of $F(t)$ is limited to $[ \log k_0 , \log k_1 ]$, it is more efficient to apply it with
\beql{eq4313}
G(t) = F(t) e^{-i \alpha t} ~,
\eeq
where
\beql{eq4314}
\alpha = \frac{1}{2} ( \log k_1 + \log k_0 ) ~.
\eeq
Then Eq.~\eqn{eq4312} yields
\beql{eq4315}
F(t) = \frac{\lambda}{\beta}  \sum_n F \lt \frac{n \pi}{\beta} \rt e^{- i \alpha ( n \pi / \beta -t)}
\frac{\sin \lambda (n \pi / \beta -t)}{ \lambda ( n \pi / \beta -t)}
h ( n \pi / \beta -t )~,
\eeq
valid for any $\beta$ and $\lambda$ that satisfy \eqn{eq436}, where we now take
\beql{eq4316}
\tau = \frac{1}{2} ( \log k_1 - \log k_2 ) ~.
\eeq
We choose
\begin{eqnarray}
\label{eq4317}
\beta & = & \pi / \delta ~, \\
\label{eq4318}
\lambda & = & ( \beta + \tau ) /2 ~, \\
\label{eq4319}
\epsilon & = & ( \beta - \tau ) /2 ~,
\end{eqnarray}
and take
\beql{eq4320}
h(u) = \frac{c}{\sinh (c)}\frac{\sinh ( c^2 - \epsilon^2 u^2 )^1/2}{( c^2 - \epsilon^2 u^2 )^1/2}~, \\
\eeq
where $c$ is a constant that was equal to 30 in most of the computations.
A typical set of values used for
the computation
of one of the
large sets of zeros near zero number $10^{20}$ is
\beql{eq4321}
\begin{array}{lll}
k_0 & = & 450 ~, \\
~~ \\ [-.1in]
k_1 & = & 1, 555, 488, 184 ~, \\
~~ \\ [-.1in]
\alpha & = & 13.637 \dd ~, \\
~~ \\ [-.1in]
\tau & = & 7.5279 \dd ~, \\
~~ \\ [-.1in]
\delta & = & 0.29 ~, \\
~~ \\ [-.1in]
\beta & = & 10.833 \dd ~, \\
~~ \\ [-.1in]
\lambda & = & 9.1804 \dd ~, \\
~~ \\ [-.1in]
\epsilon & = & 1.65258 \dd ~, \\
~~ \\ [-.1in]
c & = & 30 ~.
\end{array}
\eeq
Note that the distances between consecutive Gram points are 0.148433...,
so there is only about one grid point at which $F(t)$ is evaluated for every two
Gram intervals.

Many different kernels $h(u)$ could have been used for the interpolation.
The specific function $h(u)$ of \eqn{eq4320} was suggested by B.~F. Logan.
He had discovered a long time ago \cite{Kai} that $h(u)$ is a remarkably good approximation
to the principal eigenfunction of the finite Fourier transform, which led to its widespread
use in some signal processing applications, as well as in some problems in number theory \cite{MO}.
More important for our application are some further optimality properties of $h(u)$ that have been proved by Logan \cite{Log1,Log2}.
The formula \eqn{eq4315} is evaluated by summing the terms in the series corresponding to $n$ with $n \pi / \beta$ close to $t$,
and neglecting the remainder of the sum.
If we do not use any special knowledge of the behavior of
$F( n \pi / \beta )$ or of $\sin ( \lambda ( n \pi / \beta - t ))$,
and we sum the series in \eqn{eq4315} over $n$ with
$| n \pi / \beta - t | < c/ \epsilon$, then
we need to minimize,
$$\int_{|u| > c \epsilon^{-1}}  |h(u) u^{-1} | du ~,$$
and Logan's results show that this minimum is achieved by the function
defined in eqn{eq4320}, and equals
$$2  \log ~ \frac{1+ e^{-c}}{1-e^{-c}} ~.$$
(For $c=30$, this quantity is $\approx 2e^{-30} \approx 1.9 \times 10^{-13}$.)
Interpolation using the formula \eqn{eq4315} is performed over approximately
the interval $ [ T + c / \epsilon , T + (R-1) \delta - c / \epsilon ] $.

For the set of parameters listed in \eqn{eq4321},
the interpolating sum in \eqn{eq4315}
was estimated by explicitly evaluating and adding up about 120 terms of the sum.
Increasing $\delta$ (without changing $k_0$) increases the length of the interval
over which $F(t)$ can be computed, and therefore increases the number of zeros that can be calculated.
This has practically no effect on the running time of the rational function evaluation
program (assuming the number of grid points stays the same), but increases the time
needed by the zero-locating program, both because
there are more
zeros
to be processed, and because more terms in the interpolation formula \eqn{eq4315} have to be evaluated.
Increasing $k_0$ allows one to increase $\delta$ (and so the number of zeros that
can be computed) without changing $\epsilon$ (and thus the number of terms that have to be computed in \eqn{eq4315}).
Such a change, however, increases the running time of the zero-locating program by increasing the number of terms in the sum $Z_1 (t)$.
The choice of parameters listed in \eqn{eq4321} was not optimized carefully, and could undoubtedly be modified to obtain a more efficient algorithm.
\section{Space and time requirements}\label{sc44}
\hspace*{\parindent}
Table~4.5.1 shows the running times of the rational function evaluation
program in some of the computations that were carried out.
The first column denotes the zero set.
Upper case letters refer to the computations near the special points described
in Section~\ref{sc3} and listed in Tables~3.2.1 and 3.2.2.
Lowercase letters refer to computations listed in Table~4.6.1.
These were primarily the large sums that are described in Section~\ref{sc2},
together with some smaller computations designed to check the accuracy of the larger ones.
(See Section~\ref{sc45} for a discussion of the reasons for such computations.)
The FFT computations were fast, by comparisons, especially for $R \le 2^{19}$.
(For $R=2^{23}$ and $2^{24}$, they took several hours of elapsed time, most of it spent reading and writing disk files to rearrange the data, but only seconds of
computing times.)
The zero-locating program
took slightly under 90 minutes per million zeros when $\delta \approx 0.3$ (and less for
smaller $\delta$), so that the computation of the roughly $3.3 \times 10^7$ zeros in set $n$ took about 46 hours (2800 minutes) in addition to the 102 hours for the rational function evaluation program.

Comparison of entries $g$ and $i$, and also of $k$ and $n$, shows that increasing
the number of grid points (and therefore the number of zeros that can be computed) has
relatively little effect on the running time of the rational function evaluation program;
around the $10^{20}$-th zero, going from $1.6 \times 10^7$ zeros to
$3.2 \times 10^7$ zeros increases the running time by less than 17\%.
The reason for not using even larger grids was lack of memory.

Lack of memory, both core and disk, was the
main constraint in planning the program from the beginning.
Computing time was not a major limitation.
Around 2000 hours were used for all the computations reported here, which is substantial.
At the time these computations were carried out, however, the Cray
was
lightly utilized, and so although
only time that would have been idle otherwise was used, a lot of
it was available.
As a result, minimizing the running time of the program was not of high priority.
(Various possible improvements are discussed in Section~\ref{sc46}.)

The Cray X-MP that was used had 2 processors and 4 million words of memory
(32Mb, or megabytes).
In practice a maximum of 25Mb was available for a single process, and when such a
process ran, one processor stood stands idle.
The first version of the rational function evaluation program to be implemented had $R = 2^{19}$ and maintained all the auxiliary arrays in memory all the time, and as a result required over 15Mb.
This program was used to compute the zeros in sets $b, c$ and $e$ of Table~4.6.1, as well as the $N=10^14$ set of Table~1.2, but
it
would not run if there was any
other process of over 10Mb that was running.
For $R=2^{17}$, the corresponding program (which was used to compute all the small
sets of zeros of Section~\ref{sc3}) requires only about 5Mb, and so 
was able to utilize much more of the spare time that was available, since 
sometimes it would even run when there was
one process of $\le 20$Mb
running, and all other waiting processes were too large to fit into the remaining
memory. (This did not happen in all such cases because of the way the
scheduler was working.)
Most of the large computations were carried out with the segmented version of the program
that is described at the end of Section~\ref{sc42}.
For $R=2^{23}$, it requires 8Mb.
(This can be lowered to below 5Mb with some simple rewriting of the program.)
The main zero locating program also uses about 8Mb of space.
In this program the space requirement can be lowered to below 1Mb very easily,
since only small segments of the values of $F(t)$ at grid points are needed at any time.
The reduction of process size to 8Mb seemed sufficient, however, to take advantage of available time.

The limitation on core memory was overcome by segmenting the rational function evaluation program.
A limitation harder to overcome was the lack of disk storage space.
Most of the large computations had $R=2^{23}$, which meant that $2^{23}$ complex values of $F(t)$ were being computed and stored, which comes to 128Mb.
Moreover, this data had to be reformatted for the FFT application, so that at least for short periods, 256Mb had to be stored.
(An in-place FFT program would have eliminated the need for the extra
storage, and thus would have led to the computation of twice as many values,
but this option was not used since it would have required larger storage of final files.
This is discussed further in Section~\ref{sc46}.)
Disk space, even for temporary storage, was extremely scarce during these computations, and so this seemed to be close to the limit
of what could be easily computed at that time.
For $R=2^{24}$ (set $n$ in Tables~4.5.1 and 4.6.1),
the peak storage requirement is 512Mb, and was satisfied only 
because W.~M. Coughran kindly made available some of his dedicated disk space.

Some of the ways of overcoming the memory limitations are discussed in Section~\ref{sc461}.

Over 2000Mb of data from these computations (mostly values of $F(t)$ at 
grid points, but also some listings of zeros, as well as 
various other data) have been stored on an optical disk, and
are available for further studies. While optical disk
storage technology appears to be very reliable, some of the
data may have been corrupted in moving it over a local
area network, and so may not be usable. (When the possibility
of such errors was realized, a system of parity checks
was instituted for later data sets, to prevent such problems from arising.)
\section{Correctness of computational results}\label{sc45}
\hspace*{\parindent}
The main defect in the computations reported here is that they lack rigorous error
bounds.
This is owing to the combination of the height at which the zeta function was computed and the computer hardware that was available.
Even if one were to use the standard term-by-term evaluation of the Riemann-Siegel
formula, this problem would be severe.
The main difficulty there would be in evaluating the first sum in \eqn{eq412}, which is (neglecting the $\theta (t)$ term) of the form
\beql{eq451}
2  \sum_{k=1}^{k_1}  k^{-1/2}  \cos (t \log k ) ~.
\eeq
Near the $10^{20}$-th zero, $k_1 \approx 1.5 \times 10^9$, $t \approx 1.5 \times 10^{19}$, and so for almost all values of $k$ in the sum,
$t \log k \approx 3 \times 10^{20}$.
Therefore, since $dp$ arithmetic on the Cray is performed with about 28 decimal digits of precision, the values of $t \log k$ that are computed are accurate only to within
about
$\pm 10^{-8}$, and hence the values of $t \log k$ after reduction
modulo $2 \pi$ are only likely to be accurate to within $\pm 10^{-8}$, and the sum in \eqn{eq451} is likely to be evaluated with an error of
\beql{eq452}
E= 2 \times 10^{-8}  \sum_{k=1}^{1.5 \times 10^9}  \epsilon_k k^{-1/2} , ~~~~~-1 \le \epsilon_k \le 1 ~,
\eeq
even if the computations of the $k^{-1/2}$ and of the sum in \eqn{eq451} are performed in infinite precision.
Given no special knowledge of the $\epsilon_k$, all one can say is that
\beql{eq453}
|E| \le 2 \times 10^{-8}  \sum_{k=1}^{1.5 \times 10^9}  k^{-1/2}  \approx  1.5 \times 10^{-3}~.
\eeq
Many
examples of Lehmer's phenomenon that have been found
where the maximum of $|Z(t)| $ between zeros is substantially less than the bound \eqn{eq453},
and so in these cases one not be even certain that all the zeros are
on the critical line, much less be able to locate them accurately.

The use of multiprecision arithmetic packages would solve the above roundoff problem,
but at a high price in computing time.
In the straightforward evaluation of the Riemann-Siegel formula, one can gain a few extra digits of guaranteed accuracy by a method described at the end of Section~\ref{sc461}.
In the new method that was used for this paper, there are several additional
difficulties.
The rational function evaluation method computes the $f( \omega^h )$ as sums of a large number of terms, and some of these terms are Taylor series expansions whose coefficients are obtained by adding up numerous other expansions.
It would be a difficult task to obtain good error estimates for all these operations.
Further, even if one succeeded, it would be necessary to also bound the errors in the FFT routines and in band-limited function interpolation.
The FFT, for example, is known for its good properties in controlling errors, but
this applies to only a limited extent when one considers worst-case behavior and has to worry even about roundoff errors in addition.

The above roundoff problem would not arise if one
used
machines with larger
word sizes than the Cray's 64 bit ones,
but such computers are unlikely to become available
in the near future.
Another solution would be to restrict computations to lower heights.
However, it seemed desirable to obtain information from as high up as possible,
since the zeta function approaches its asymptotic
behavior slowly.
Therefore it was necessary to abandon rigor in the computations.

While no rigorous error bounds have been obtained, the computational results are thought to be accurate.
One reason for thinking this is based on heuristics.
The bound \eqn{eq453} is very conservative in that it is sharp only when almost
all the $\epsilon_k$ in \eqn{eq452} are close to $+1$ or almost all are close to $-1$.
In practice, one expects that the $\epsilon_k$ will be practically independent of each other, and if that is so,
then even under the assumption that the $\epsilon_k$ take only the extreme value $\pm 1$, we find that the rms value of $E$ is only
$$2 \times 10^{-8} \lt \sum_{k=1}^{1.5 \times 10^9} k^{-1} \rt^{1/2}  \approx  9 \times 10^{-8} ~.$$
This is the typical error we expect
as a result of
cancellation among various roundoff errors.
Similarly, one expects substantial cancellation in the rational function evaluation, in
the FFT, and in band-limited function interpolation.

The need to rely on cancellation of roundoff errors introduces another level of uncertainty to the computation.
Although it is common and accepted in numerical analysis, statistics, and the physical sciences,
it is seldom encountered in pure mathematics.
This uncertainly is added to the usual uncertainties about reliability of hardware,
the design of hardware floating point units ({\em cf.}~\cite{Br,Od2,Schr1,Schr2}), the correctness
of manufacturers' mathematical subroutines ({\em cf.}~\cite{SF}), the reliability of compilers, and finally the correctness of the main programs.
All these problems occur with reasonably high frequency.
To add to the long list of problems that have been found, we mention that with some Cray Fortran compilers, the test program for the Brent MP package \cite{Br4}, which computes $\pi$,
$\exp ( \pi (163/9)^{1/2} )$, and $\exp ( \pi (163)^{1/2} )$ to about 100 decimal places, produced all the digits of $\pi$ correctly,
but gave it a negative sign, and produced totally unrecognizable numbers for the remaining two problems.
Mathematicians usually insist on a higher standard of rigor than this.

In some mathematical computations it is not important to have absolute assurance of correctness, since the results are used only to obtain insight into behavior of various functions or systems, and eventually conventional proofs that make no appeal to any computations are constructed ({\em cf.}~\cite{Od0}).
In many other cases, though, such as that of the Four Color Theorem \cite{AH},
or of some proofs in dynamical systems ({\em cf.}~\cite{Lan}), computational results are an integral part of the proof.
There is a school of thought that questions the validity of all such proofs.

The computations of this paper are in one sense even more questionable than those mentioned above, since they depend not only on the correctness of the hardware and software, but also on quasi-random cancellation of roundoff errors.
This is to some extent worse than relying on usual probabilistic algorithms, since in these at least the coin tosses are really independent, so one can talk of rigorous probabilistic results.
(This assumes, of course, that one can obtain really random bits, but that is another topic we will not deal with it here.)
In our case there is no true randomness, as the roundoff process is deterministic.
Moreover, the zeta function is certainly nonrandom, and so it is certainly conceivable that the errors in the evaluations of $Z(t)$ might
arrange themselves to conceal a
violation of the RH.
It is for this reason that the previous numerical verifications of the RH, such as those of
Brent \cite{Br5} and of van~de~Lune, te~Riele, and Winter \cite{LRW2},
were done very carefully.
For example, those investigators
did not even rely on their machines' cosine routines, and were careful in the analysis of their error terms.
As a result, the validity of the verification of the RH for the first $1.5 \times 10^9$ zeros by van~de~Lune et~al. relies only on the assumptions that the hardware and compilers were reliable, their program was correct (it is available for inspection in \cite{LRW1}
and some further modifications are in \cite{WR}), and that their machines' $dp$ cosine routines
(used to provide data to the linear interpolation routines that compute cosines)
were at least moderately accurate.

The new programs do not have the same assurance of correctness that those of \cite{LRW2} do.
However, in a sense they can be argued to be even more trustworthy.
The reason for this is that large parts of the computations were done twice.
In general, redoing the same computation on the same machine with the same program provides a check only against certain intermittent errors.
In our case, though, the computations were quite different.
The grids $T$, $T+ \delta$, $T+ 2 \delta , \dd$, at which $F(t)$ was being evaluated were always different.
As a result, the rational functions $f(z)$ that were being evaluated at the $R$-th roots of
unity (where $R$ was sometimes the same and sometimes different in different computations) were different for the two grids.
Therefore the numbers that resulted from the application of the FFT, and were used for band-limited function interpolation, were different.
That what was being computed in the two calculations was
in both cases
$Z(t)$ was thus not apparent at all from the numbers being processed, and is a result of the involved
analysis of Sections~\ref{sc41} to \ref{sc43}.
That the two values that were obtained were the same to within the
expected error serves as evidence that they are indeed values of $Z(t)$,
since it would require a very unusual combination of errors for the two computations to yield the same answers otherwise.
This method thus serves to check not only the roundoff errors, but also the hardware, compilers, operating systems, and the program themselves.

Care was taken to minimize the parts of the computations of $Z(t)$ that were common to the overlapping sets.
The values of $\delta$ were always different.
The computations of $Z_1 (t)$ and of the asymptotic expansion in the Riemann-Siegel formula were harder to make distinct.
However, the procedure for computing $\theta (t)$ outlined in Section~\ref{sc41} served to make the values of $\theta (t)$ in different computations slightly different, so that in locating zeros, different computations dealt with different values of $t$.

The above method of computing $Z(t)$ in two different ways that ought to yield the same result only because of deep mathematical results is not novel.
In early computations of $\pi$, such as that of Shanks and Wrench \cite{SW} (see \cite{BB} for history of this subject and much more efficient modern methods), $\pi$ was calculated
through two different Machin-like formulas.
In the case of \cite{SW}, they were \\

\noindent
$\begin{array}{l@{~~~~~~~~~~~~~~~~~~~~}l@{~}l@{~}l}
& \pi & = & 24 ~\mbox{arctan}~ \lt \df{1}{8} \rt + 8 ~\mbox{arctan}~ \lt \df{1}{57} \rt + 4 ~\mbox{arctan}~ \lt \df{1}{239} \rt \\
~~ \\ [-.1in]
\mbox{and} \\
~~ \\ [-.1in]
& \pi & = & 48 \mbox{arctan}~ \lt \df{1}{18} \rt + 32 ~\mbox{arctan}~ \lt \df{1}{57} \rt - 20~\mbox{arctan}~ \lt \df{1}{239} \rt ~.
\end{array}$ \\

\noindent
Since again only a very unusual combination of errors could give the same answer by both methods,
obtaining the same result
provides a convincing (although nonrigorous) argument in favor of correctness.

With new algorithm, the comparison of
results of overlapping computations did lead to the uncovering of one error in the program that had escaped detection in several earlier runs.
After one computation of over $1.6 \times 10^7$ zeros near zero number $10^{20}$ (set $m$ of Table~4.6.1, described below), the set below it (set $l$ of Table~4.6.1) was computed.
However, a computation of the zeros in the segment overlapping set $m$ showed apparent violations of the RH when the values of $F(t)$ from set $l$ were being used, although no such violations were found using set $m$.
Interpolation of values of $F(t)$ from set $m$ to give the values of $F(t)$ on the grid of set $l$ revealed that the computed
values of $F(t)$ at the grid points $f_j = T + j \delta$ of set $l$ differed from the (presumably correct) ones derived from set $m$ by
$c(-1)^j$, where $c$ was a certain constant.
This immediately suggested that in set $l$, $f(-1)$ was being evaluated incorrectly.
An inspection of the code revealed a simple mistake
having to do with indexing of the roots of unity $\omega^h$ in the 
segmented program, and it was easy to correct the data.
This bug had not revealed itself before because the unusual combination of having a
pole of the rational function $f(z)$ in a certain range
close to $-1$, which was required for the
code to produce incorrect output,
had not occurred in the earlier runs.

How close to each other are the values of $Z(t)$ computed
in different runs?
Let us consider the neighborhood of the extreme example of
Lehmer's phenomenon
near $\gamma_n$, $n=10^{18} + 12, 376, 780$, where the minimum of $Z(t)$ between $\gamma_n$ and $\gamma_{n+1}$ is only $-5.3 \times 10^{-7}$.
This is the example that comes closest to violating
the RH among all those found in our computations,
and the obvious question is whether one can be sure that the RH is indeed satisfied
by $rho_n$ and $rho_{n+1}$.
This example was found in a computation of $1.7 \times 10^7$ zeros,
and to confirm the accuracy of the computed values of $Z(t)$, two additional
computations were carried out, each of about $1.5 \times 10^5$ zeros,
and each centered close to $\gamma_n$.
The starting points of the three computations and the grid spacings $\delta$ were
distinct in all three computations to assure
maximal independence in the computed values.
When the results of the these runs are plotted on the scale of Fig.~2.7.1, they
are indistinguishable.
When one plots $Z(t)$ only in the immediate vicinity of $\gamma_n$ and $\gamma_{n+1}$, as in Fig.~4.6.1, the three graphs are still indistinguishable.
It is only when one goes over to the scale of Fig.~4.6.2,
which shows $Z(t)$ near its minimal value in $( \gamma_n , \gamma_{n+1} )$,
that differences are apparent.
This graph was prepared by computing $Z(t)$ from each run at intervals of $10^{-6}$ times the length of a Gram interval (so that Fig.~4.6.2 corresponds to about 150 evenly spaced values of $t$)
and connecting the points obtained that way by lines.
(The function of the lines was to enable the reader to tell which values come from the same data set.)
The jagged appearance of the lines is the result of the quantization and roundoff
errors.
(Note that changing a value of $t \approx 1.7 \times 10^{17}$ by $10^{-6}$ times the length of a Gram interval
affects only the last 15 or so bits in the $dp$ representation of $t$.)
Given the scale of Fig.~4.6.2, the three curves are close together,
and thus provide convincing evidence that the claimed values
of $Z(t)$, $\gamma_n$, and $\gamma_{n+1}$ are indeed highly accurate, and that the RH is not violated in this region.

All indications from preliminary runs were that the new algorithm was highly accurate,
and storage of two complete data sets needed to perform
a detailed comparison would have been hard to arrange.
Therefore it
was decided not to
recompute all the values of zeros near zero number $10^{20}$ using different grids,
but to have different computations cover consecutive ranges with some overlap.
Table~4.6.1 shows all the different sets of zeros that were computed and that overlapped other ranges.
The three sets of zeros that were referred to above in the discussion of Lehmer's
phenomenon, for example, are listed under $f, g,$ and $h$ in Table~4.6.1.
(The set $a$ consists of the zeros computed in \cite{Od2} by the standard
Riemann-Siegel formula method, and so its values of zeros are very trustworthy.)
The four main computations (near $N=10^{20}$) are those in sets $k, l, m$, and $n$, and each one overlaps each of its neighbors in about $10^6$ zeros.
The small set o was computed as an additional check, since the smaller grid spacing and fewer grid points were expected to produce more accurate values,
and the somewhat different program used was an extra check on programming mistakes.
(This was also the motivation behind some of the other computations of small sets of zeros,
such as that of $j$.
The medium size sets, $b, c$ and $e$, were computed by the earliest of all versions of the new program.)

A few large scale statistical comparisons were made of the values of $Z(t)$ produced in different computations.
For example, to compare sets $m$ and $o$
of Table~4.6.1,
the values of $Z(t)$ were computed using data from each set at
$5 \times 10^6$ points spaced 1/300 apart (about 45 per Gram interval) starting at $t = 1.52024401159207401 \times 10^{19}$.
The largest difference (in absolute value) was $1.5 \times 10^{-6}$,
and the rms difference was $5 \times 10^{-8}$.

While the errors made in computing $Z(t)$ are of some interest, the main question
is that of accuracy in computing the $\gamma_n$,
which depends not only on accuracy of values of $Z(t)$, but on the size of $Z(t)$ and
$Z' (t)$ near zeros.
Therefore extensive and careful comparisons were made of the differences
in values of $\gamma_n$ computed in different sets.
Table~4.6.2 summarizes the results of these comparisons.
The ``$a$ vs. $b$'' entry, for example, shows that the values for the 101,053 zeros common to
sets $a$ and $b$ different by no more than $2.5 \times 10^{-9}$,
and the rms difference was $3.7 \times 10^{-1}1$.
(These are differences in the values of the $\gamma_n$.
Should the values of two adjacent zeros in set $l$, for example,
each be off by $\epsilon$, with one value too small by $\epsilon$
and the other too large by $\epsilon$, the resulting value of
$\delta_n$ would be off by $\pm 14 \epsilon$.)
The maximal differences increase as one looks down the table,
as was to be expected.
They all stay small, though, and are the main justification
for the claimed validity of the data.

The rms difference entries in Table~4.6.2 should be treated
with great caution.
One reason is that the zero-locating program was only asked
to compute the zeros to a nominal accuracy of $\pm 2 \times 10^{-8}$
(for zeros near zero number $10^{20}$; somewhat higher accuracy
was specified for lower zeros).
Because of the mixture of linear and quadratic interpolation
that was used, usually the convergence of the algorithm at the
end of a particular search was quadratic, and so accuracy much
greater than the specified one was reached in almost all cases.
Thus the fact that the rms figures in Table~4.6.2 are
substantially below the specified accuracy of $2 \times 10^{-8}$ is
the result of many happy accidents, and not a matter of
design.
Another reason not to rely on the rms figures is that
often
they were inflated by the programs that were used for the comparisons.
Since there was no reason to expect accuracy better than $\pm 10^{-8}$, $sp$
programs were used for most of the computations of Table~4.6.2, which
led to the loss of the last few bits of precision.
(The anomalously large rms value for the ``$l$ vs. $m$''
entry, as compared to the ``$k$ vs. $l$'' and ``$m$ vs. $n$''
entries, which cover roughly the same number of zeros at about
the same height,
is almost certainly due to the use of a $s p$ array in a data
conversion routine, for example.)
Thus in general the rms figures in Table~4.6.2 are upper bounds for the
rms errors achievable with the new algorithm, but should not be
regarded as accurate estimates.

One source of errors in the computation of the $\gamma_n$
lies in the method of calculating $\theta (t)$.
Some of these errors come from the
roundoff difficulties associated with handling large numbers
within the limited precision that was available.
Other errors came from the Taylor series expansion procedure,
described in Section~\ref{sc41}, that was used to compute $\theta (t)$.
Some indication of the errors introduced this way can
be obtained from the data produced during the main runs.
Because of the inefficient search procedure near exception to
Rosser's rule (described in Section~\ref{sc41}), usually about
ten zeros near each exception were computed twice.
The two values were hardly ever identical, since the zero locating program
was usually invoked with different arguments.
Differences caused by this factor were usually extremely
small.
Much larger were differences caused by the fact that
often the two computations calculated $\theta (t)$ for nearby
values of $t$ by expanding around different values of $t_0$.
The largest difference in the computed values of the $\gamma_n$
that was found that is due to this phenomenon is $2.7 \times 10^{-7}$
for $n= 10^{20} + 31, 141, 844$.
(The second largest was only slightly more than half as large.)
This zero is located near two exceptions to Rosser's rule that
are close to each other, with the peak value of $Z(t)$ in
that region equal to 257.6.
The pattern of zeros (starting at Gram point $g_{n-19}$)
is 2111110110030101311, and $Z(t)$
is small in a large neighborhood of
$\gamma_n$ ($| Z(t) | < 0.015$ over approximately the
whole Gram interval that contains $\gamma_n$).
$Z' ( \gamma_n) = 0.7$ is small, and so the
computed location of $\gamma_n$ is sensitive to errors in
the computation of $\theta (t)$.

Other tests to determine the sensitivity of the computed values of the zeros
to errors in the computation of $\theta (t)$ were also performed.
For example, the zeros in set $o$ were computed several times, always using
the same rational function evaluation output for the interpolation
of band-limited functions, but modifying the strategy of evaluating
$\theta (t)$ by forcing more frequent recomputations of $t_0$, or
simply the use of different sets of $t_0$'s.
The resulting values for the zeros had differences
(when compared to the basic computation of the zeros
in that set) that were $\le 10^{-7}$ in absolute value,
and $\le 2 \times 10^{-9}$ in rms value.
Thus the basic conclusion from these tests is that errors in the
computed $\theta (t)$ were not a significant factor.

Another reason for trusting the computational results of this paper is that the results of the most time-consuming part, the rational function evaluation, are transformed by the FFT before
being used for the computation of zeros.
This means that any error in this part of the computation affects the computation of all zeros, and so if it is substantial, is likely to
lead to
an apparent
counterexample to the RH.
This is in contrast to the standard methods, such as that of \cite{LRW2}, in which a single mistake affects the computation of only one value of $Z(t)$.

The final, and in
many ways most
convincing, although
unrigorous argument in favor of the
correctness of the computations reported here is that they did not
find any counterexamples to the RH.
This might seem a strange argument.
The point of it is that if the RH is true, it is only barely true, in the sense that even tiny changes in the formulas used to compute the zeta function yield functions that no longer satisfy the RH. Many deliberate as well as accidental
experiments were performed in which some of the parameters in the
programs were modified slightly, and they almost invariably
ended up giving apparent counterexamples to the RH.
For example, in set $V$ of Section~\ref{sc3}, the minimal
$w_n$ (see Section~\ref{sc27} for definitions) that was found was $1.43 \times 10^{-4}$, so perturbing $Z(t)$ by smaller quantities could not produce apparent
counterexamples to the RH.
In particular, dropping the asymptotic expansion part of the Riemann-Siegel
formula does not produce visible problems in this set, although it does in other ones that have
more extreme cases of Lehmer's phenomenon, and in all cases it
perturbs the computed values of the zeros.
On the other hand, only slightly larger perturbations do produce apparent counterexamples.
One also finds counterexamples when one computes
$$Z(t) - 2k^{-1/2} \cos ( t \log k - \theta (t) )$$
for $k=10^6$.
Also, when one computes
$$Z_1 (t) + \mbox{Re} ~ e^{- \theta (t)} ~ F(t- \alpha )$$
with $\alpha = 10^{-4}$ instead of $\alpha =0$, apparent counterexamples
to the RH appear.
(In all these cases, the apparent counterexamples refer to cases where the function
being computed has a positive relative minimum or a negative relative maximum.)
\section{Possible improvements}\label{sc46}
\hspace*{\parindent}
At large heights, the new algorithm is much faster than previous methods.
The computation of $10^5$ zeros near zero number $10^{12}$ in \cite{Od2}
took about 15 hours on a Cray X-MP using direct evaluation of the Riemann-Siegel formula.
Set $n$ of Table~4.6.1 contains almost $3.3 \times 10^7$ zeros near zero number $10^{20}$, and it was computed in about 150 hours on the same machine.
Since the Riemann-Siegel formula involves about $7.5 \times 10^3$ times more terms near zero number
$10^{20}$ than near zero number $10^{12}$,
computing all the zeros in set $n$ by the method of \cite{Od2} would have
required about
$$15 \times 300 \times 7500 \approx 3.7 \times 10^7$$
hours, or more than $2 \times 10^5$ times longer than the new algorithm required.

While the current implementation of the new algorithm is much
more efficient than previous algorithms, it is far from optimal.
The author's main interest was in demonstrating that the new algorithm was indeed faster than old ones, and in obtaining data about zeros of the zeta function.
Since spare computer time was available, saving programming effort was often chosen over
efficiency of the program.
The following subsections present some of the ways in which the program could be modified to run faster or to produce more accurate results.
They might be useful in future computations.
It seems likely that the ideas in Sections~\ref{sc461} and \ref{sc462} could be used to increase the speed of the algorithm by another order of magnitude on the
Cray X-MP.
This might make it possible to compute large sets of zeros near zero number $10^{22}$, for example.

All the main programs can be parallelized, and one can achieve high performance this way.
(For the rational function evaluation program, there are some examples of similar algorithms, discussed in Section~\ref{sc462}, that have been implemented effectively on parallel computers by
Greengard and Gropp \cite{GG},
Zhao \cite{Zh}, and Zhao and Johnsson \cite{ZJ}.)
One difficulty
in using existing multiprocessors would likely be their relatively low precision.
Our discussion will be oriented towards more standard vector processors, however.
\subsection{Faster and more accurate computations}\label{sc461}
\hspace*{\parindent}
Many parts of the rational function evaluation
program can be speeded up.
Table~4.3.1 shows that the present strategy of computing $dp$ values of $\log k$ takes about 10\% of the
running time.
While this is 1.5 times faster than using the current standard
Cray $dp$ routines would be (and 6 times faster than using the old Cray routines), it can be
improved substantially by modifying the program slightly.
For example, the following method is about 3.5 faster (for $k \approx 10^9$) than the one currently used.
Group the $k \in S_{m,j}$ into consecutive blocks, say $k_4 \le k \le k_5$,
with $k_5 - k_4 \precsim 2 \times k_4 \times 10^{-6}$,
and let $k_6 = \left \lf ( k_4 + k_5 ) /2 \right \rf$.
Compute the $dp$ $\log k_6$ using the Cray routine, and compute
the $h_j = 1/(j \cdot k_6^j )$ in $dp$ for $1 \le j \le 4$.
Initially, for $k_4 \le k \le k_5$, assign to $d_k$ the value of $\log k_6$, and then, for $1 \le j \le 4$, modify each $d_k$, $k_4 \le k \le k_5$, by
subtracting from it $h_j (k- k_6 )^j$, where the
$(k-k_6 )^j$ are taken from a precomputed integer array.
For $j=4$, the multiplication of $h_j$ and $(k-k_6 )^j$ can be carried out in $sp$.
Further improvements can probably be obtained with further experimentation.
(The speeds that can be achieved in this part of the program depend strongly on how the compiler treats $dp$ computations.)

Substantial savings can be obtained by modifying the procedures used to evaluate the $f_{p,q} ( \omega^h )$.
Instead of choosing a uniform threshold $Q_1$, the decision whether to use Taylor series expansions
or direct evaluation can be made dependent on the size of $I_{p,q}$.
Reducing the number of terms in the Taylor series as mentioned in Section~\ref{sc42} can reduce the total running time by at least 15\%, and even greater savings are probably possible by more careful choices.

The zero-locating program can be improved in several ways.
The computation of $\theta (t)$ takes about 7\% of the time for the choice of parameters in \eqn{eq4321}.
This time can be reduced practically to zero, since around the $10^{20}$-th zero, $\theta (t)$ is almost linear
and the distances between consecutive Gram points are almost constant.

The strategy for locating zeros can be improved, especially in the neighborhoods of exceptions to Rosser's rule, where it is grossly inefficient.
Currently about 8.5 evaluations of $Z(t)$ are used to compute each zero (with one more evaluation to obtain the value of $Z(t)$ at the midpoint between adjacent zeros).
It should be possible to devise strategies that take better advantage of the previously computed values and of the expected
behavior of $Z(t)$.
This might involve computing $Z' (t)$ (which can be obtained from the
interpolation formula \eqn{eq4315}) and modifying
Brent's algorithm \cite{Br1}.
A good model for this approach is the algorithm of van~de~Lune et~al. \cite{LRW1,LRW2},
which uses fewer than 1.2 evaluations of $Z(t)$ per zero to separate the zeros.

About
45\% of the time of the zero-locating program (for the parameters listed in \eqn{eq4321}) is spent in band-limited function interpolation.
(The computation of $Z_1 (t)$ takes approximately 25\%.)
Logan's kernel \eqn{eq4320} has desirable optimality properties in terms of rate of convergence of the interpolating sum \eqn{eq4315},
but it is somewhat hard to compute.
It is possible that other kernels could be constructed that would give slower convergence, and therefore require evaluating explicitly more terms in the sum in \eqn{eq4315}, but which would be much more efficient to compute.
Another approach would be to initially use the formula \eqn{eq4315} only for values of $t$ that belong to a grid somewhat finer than that of
$T$, $T+ \delta$, $T+ 2 \delta , \dd$,
say at points $T+ k \delta / 1000$, $k \ge 0$.
For such values of $t$, it would only be necessary to precompute the
values of
$$
e^{i \alpha m \delta / 1000} \frac{\sin ( \lambda m \delta / 1000 )}{( \lambda m \delta / 10000 )}  h( m \delta / 1000 )
\eqno{(4.7.1.1)}
$$
for $|m| \precsim 1000 c/( \delta \epsilon )$, and later evaluations of the interpolation formula would be reduced to inner products of vectors.
With this approach, even without the use of assembly language, one could compute
interpolation sums of the form \eqn{eq4315} at the rate of about $1.2 \times 10^7$
terms in the sum
per second on the Cray X-MP.
(If one selects $\delta$ to be a simple rational multiple of the average gap between Gram points, say
$\delta = 2( g_{n+1} - g_n )$, another factor of 2 improvement in speed is
possible by taking advantage of the fact that only $\mbox{Re} ~e^{- i \theta (t)} F(t)$ is needed, and $\theta (t)$ is close to linear.)
In the existing program, they are computed at the rate of only about $5 \times 10^5$ terms per second.
Once the zero was proved to lie between two points of the subgrid
$T+ k \delta / 1000$, one could locate it more accurately by evaluating $Z(t)$ at several neighboring points of the subgrid, using some numerical interpolation method to obtain an approximation to the zero that is likely to be accurate.
Finally this location of the zero could be confirmed by using the standard
interpolation method to evaluate $Z(t)$ at points on either side of the zero.

All the main programs are written in Fortran, and many parts of the computation are handled by subroutines.
Some slight improvements can be expected from replacing subroutine calls,
which are slow on the Cray machines, by in-line code.
Much greater speedups are likely to be achieved by using assembly language.
Currently none of the $dp$ operations vectorize.
However, given the structure in the $dp$ operations employed in the new algorithms, it should be possible to write assembly language code that would vectorize these operations.
Another case where assembly language ought to produce much faster programs is in the computation of Taylor series coefficients and the evaluation of Taylor series,
which, as is shown by Table~4.3.1, account for more than half of the running time of the rational function evaluation program.
Let $c_n$ be defined by
$$\sum_{k \in S_m,j} \frac{a_k}{z- b_k} = \sum_{n=0}^\In  c_n ( z_{p,q} - z )^n ~,$$
so that
$$c_n = \sum_{k \in S_m,j}  a_k ( z_{p,q} - b_k )^{-n-1} ~,$$
and the $c_n$ corresponding to different sets $S_{m,j}$ are added together to obtain the $n$-th Taylor series coefficient of $f_{p,q} (z)$.
The
$c_n$ are computed in the present program
by using
two complex $sp$ arrays,
$u_k$ and $v_k$, $k \in S_{m,j}$, with
\begin{eqnarray*}
u_k & = & (z_{p,q} - b_k )^{-1}~, \\
v_k & = & a_k ( z_{p,q} - b_k )^{-n} ~.
\end{eqnarray*}
To compute $c_n$, $v_k$ is assigned the value $u_k v_k$ for $k \in S_{m,j}$, and the Cray library function $c s u m$ is
invoked to sum the new $u_k$.
($C s u m$ sums a complex $sp$ array in a vectorized way.)
Each $c_n$ thus requires $| S_{m,j} | $ complex $sp$ multiplications and
$| S_{m,j} | - 1$ complex $sp$ additions.
These combinations of complex multiplications and additions are carried out at the rate of $1.29 \times 10^7$ per seconds.
Since each complex multiplication involves 4 real multiplications and 2 real additions, and each complex addition requires 2 real additions, the Cray is
performing $1.03 \times 10^8$ floating point operations per seconds,
which is good when one recalls that the cycle time of the Cray X-MP is 9.5 nanoseconds.
However, it should be possible to take advantage of the fact that each of the basic combinations
of a complex multiplication and complex addition involves 4 real multiplications and 4 real additions.
Since the Cray can do an addition and a multiplication at the same time,
it ought to be feasible to write the code so that additions coming from computation of some particular $n$ would be done at the same time as the multiplications for $n+1$, say.
This would give a speedup factor 2, if the data transfers could be arranged appropriately.
(It might even be possible to obtain some savings in Fortran, without resorting to
assembly language.)
Similar improvements can probably be obtained in other parts of the program.

Table~4.3.1 shows that many different parts of the rational function evaluation program consume noticeable fractions of computing time, and so for maximal efficiency one would have to work on all of them.
This is also true of the zero-locating program.
No matter to what extent these programs are optimized, however, one can obtain some savings by optimizing the choice of the
parameters $\delta , k_0$, etc.,
which were not chosen very carefully in the computations that are reported here.

Memory constraints can be overcome by using larger mass storage devices.
These could be larger magnetic disks (the maximal total storage requirement
for computing the $3.3 \times 10^7$ zeros in set $n$ was 512Mb, and much larger
disks are commercially available), or even magnetic tapes (especially the high capacity digital tapes that are becoming available) or optical disks.
Substantial savings can be realized by using in-place FFT programs, which perform the FFT on a set of data with
little additional space being required.
Most such algorithms require
about
$\log R$ passes through the data for an
$R$-point FFT.
D.~H. Bailey has pointed out that there is an algorithm of Gentleman \cite{Gen} which requires only three passes through
the data, and so is particularly attractive for our application.

One factor that might facilitate very large scale computation with the new algorithm is that most of computing
time of the rational function evaluation program is spent in only a few segments,
because of the
nonuniform distribution of the $\beta_k$.
This means that one can first perform most of the computation with very little storage, and only a couple of hours would be needed
to deal with the overwhelming majority of segments and the FFT program, and it's only
during that time that substantial storage would be needed.
Afterwards, the data can be stored away even on slow mass storage devices, since the zero-locating program requires only small segments of data at a time.
Furthermore, storage space can sometimes be used more fully by choosing $R$ not
to be a power of 2.

At the time the program was implemented, limitations on disk storage, capacity of local area networks, and availability of long-term storage on optical
disks were such that utilizing the methods suggested above seemed very cumbersome.
Right now, however, with the availability of an automatic optical disk changer and larger disks, it would be
much easier
to carry out some of these improvements.

It is possible to gain some additional speed by taking advantage of the nonuniform distribution of the $\beta_k$.
Instead of computing $F(k_0 , k_1 , t)$ at a grid of point $t = T, T+ \delta , \dd , T + ( R-1) \delta$, one can compute
$F(m_j , m_{m+1} ; t)$ for $0 \le j < s$, where
$$m_0 = k_0 < m_1 < \cdots < m_s = k_1 ~,$$
where the $m_{j+1} /m_j$ are roughly equal, and where $t$ now would run over a much
sparser grid, approximately $t = T$, $T+ s \delta$, $T+ 2 s \delta , \dd $, since the range of
frequencies in each $F( m_j , m_{j+1} ; t)$ would be much narrower.
In order not to use too much space, one could then compute
$F(m_j , m_{j+1} ; t)$ at only $R/2^{s-j}$ points at a time.
This would mean that the $F(m_j , m_{j+1} ; t)$ for $j \le s-2$ would be computed
on several adjacent grids of points, but since in the ranges that are being considered now, the running time depends mostly on $k_1$, and to a much smaller
extent on $R$, it appears that one could obtain substantial savings.

It should be possible to obtain slightly more accurate results at very small
additional cost in computing time.
This might be useful if one were to do some computing near zero number $10^{22}$, for example.
The main source of inaccuracy in the present program is in computing $T \log k$ mod $2 \pi$, $2 \le k \le k_1$.
While multiprecision programs such as Brent's MP package \cite{Br4} are likely to be too slow to be used on each such term separately, one can apply them for some values of
$k$ spaced far apart, and then use Taylor series expansions in terms of $k$
to obtain the other values.
One has
$$T \log (k+h) = T \log k + Th /k -
Th^2 / ( 2k^2 ) + \cdots ~,
$$
and so if $| h/k | \le 10^{-3}$, say, $dp$ arithmetic would give
about 4 more decimal digits of accuracy than the method that is now used.
\subsection{Greengard-Rokhlin algorithm}\label{sc462}
\hspace*{\parindent}
The possible modifications to the present implementation that are discussed in Section~\ref{sc461} are small programming improvements.
It is also possible to change the basic rational function evaluation algorithm, by
modifying the functions $f_{p,q} (z)$ (see Section~\ref{sc42}).
What is needed is a collection of functions $\tilde{f}_j (z)$ such that for every $h$,
$$f( \omega^h ) = \sum_{j \in J(h)}  \tilde{f}_j ( \omega^h )$$
for some subset $J (h)$, and such that all the $\tilde{f}_j ( \omega^h )$ can be computed efficiently by direct evaluation or by using their Taylor series expansions.
The functions $f_{p,q} (z)$ of \cite{OS} and of Section~\ref{sc42} are only one of many choices.

As was already noted in \cite{OS} in remarks about the Trummer problem, the rational
function evaluation algorithm of that paper can be extended to the
evaluation of much more general functions.
Another algorithm for the evaluation of Coulomb and gravitational potentials was 
invented by Greengard and Rokhlin \cite{GR1}, and was subsequently 
improved, extended,and applied to several additional problems by several investigators \cite{AGR,CGR,GG,GR2,GR3,Kat,Zh,ZJ}.
It seems to offer the possibility of a substantial improvement in the speed of
the zeta function program.
Its underlying principle is the same as that of the algorithm of \cite{OS},
namely of aggregating the contributions of those poles of the function that are close together.
However, it works differently.
To avoid unnecessary
notation, we explain briefly how it would be applied to the zeta function
problem, which is the evaluation of $ f( \omega^h ) $, although
it is more general than that.
In the algorithm of \cite{OS} and this book, 
the functions $ \tilde{f}_j (z) $ are usually evaluated
by obtaining their Taylor series expansions around points
outside the regions containing their poles.
For each $m$, and each $k \in S_m$, the coefficients $a_k ( z_{p,q} - b_k )^{-n-1}$ in \eqn{eq4215} are evaluated for $0 \le n \le V$
($V$ is usually around 40) and for each pair $p, q$ such that $S_m \subseteq I_{p,q}$,
$Q_1 \le q \le Q$.
Since there are
about
$Q \approx \log R$ such pairs $p,q$, the total
effort involves on the order of
$$
Vk_1 \log R
\eqno{(4.7.2.1)}
$$
basic arithmetic operations.

In the Greengard-Rokhlin algorithm, one would compute
instead the coefficients $a_k ( b_k - z_m )^n$ in the expansions
$$
\frac{a_k}{z - b_k} = \sum_{n=0}^\In  a_k ( b- z_m )^n ( z - z_m )^{-n-1}~,
\eqno{(4.7.2.2)}
$$
for example.
Here $z_m$ would be some point located among the $b_k$ with $k \in S_m$,
say
$$z_m = \exp ( 2 \pi i ( m+ 1/2 ) /R ) ~.$$
Expansions of the type (4.7.2.2) converge for $z$ away from $z_m$ and $b_k$, say for
$| z - z_m | \ge 3/2$.
The coefficients $a_k ( b_k - z_m )^n$ would again be computed for $0 \le n \le V$, with $V$ of about the same
size
as in the
present algorithm.
Addition of the coefficients $a_k ( b_k - z_m )^n$ for $k \in S_m$ gives
an expansion
$$
\sum_{k \in S_m} \frac{a_k}{z- b_k} = \sum_{n=0}^V  A_n^{(m)} ( z - z_m )^{-n-1} + \cdots
\eqno{(4.7.2.3)}
$$
that can be used to compute the contribution of the $k \in S_m$ at $z= \omega^h$ for $\omega^h$ close to $z_m$.
For $\omega^h$ that are further away, one would combine the contributions of several $S_l$'s, say $S_m$, $S_{m+1}$, $S_{m+2}$, and obtain an
expansion around $z_{m+1}$.
The crucial point about the Greengard-Rokhlin algorithm is that unlike in the algorithm of \cite{OS} and this book,
this expansion around $z_{m+1}$ would not be done by recomputing the contribution
of each $k \in S_m cup S_{m+1} cup S_{m+2}$, but by
translating the previously computed expansions; e.g.,
$$
\sum_{n=0}^V  A_n^{(m)} (z-z_m )^{-n-1} =
\sum_{n=0}^V ~ B_n^{(m)} (z-z_{m+1} )^{-n-1} + \cdots ~,
\eqno{(4.7.2.4)}
$$
where the $B_n^{(m)}$ are derived from the $A_n^{(m)}$ by linear transformations coming from the binomial expansion, without reference to the $a_k$ and $b_k$ for $k \in S_m$.
The straightforward formulas for the $B_n^{(m)}$ take on the order of $V^2$ operations to compute them from the $A_n^{(m)}$.
As a result, since there would again be on the order of $\log R$
levels in the hierarchy of expansions, with each level having only 1/3 or 1/2 of the
expansions in the level below, obtaining all the expansion coefficients would take on the order of
$$
k_1 V + V^2 R
\eqno{(4.7.2.5)}
$$
operations.
For $k_1 \approx R$ and $V \approx \log R$, as would be
true for the zeta function algorithm in the absence of memory
constraints, and
also
for the Coulomb or gravitational potential calculations of
\cite{GR1} and related papers,
this is about the same operation count as for the present algorithm (see (4.7.2.1)).
However, for present and foreseeable computations of the zeta function at large heights, $R$ is much smaller than $k_1$
$(R \approx 1.6 \times 10^7$ as compared to $k_1 \approx 1.5 \times 10^9$ in the largest computation of this paper), and so the Greengard-Rokhlin algorithm is likely to give much faster rational function evaluation.
An order of magnitude improvement
seems likely in
the time needed to evaluate the expansion
coefficients.
At present, Taylor series expansions consume about half of the time, so even eliminating them entirely would only double the speed of the program.
However, with faster coefficient expansion techniques, one could also
use these methods to evaluate contributions to $f( \omega^h )$ of $b_k$
that are closer to $\omega^h$ than is done at present, and this would give much greater gains in efficiency.
Moreover, once these parts of the program were improved, the improvements to other parts (such as that of evaluating $dp$ values of $\log k$) that have been suggested
would become much more significant, and all of them together could increase the speed of the entire algorithm by an order of magnitude,
especially if assembly language was used as suggested in Section~\ref{sc461}.

The basic idea of translating an expansion that is at the heart of the
Greengard-Rokhlin algorithm for pole expansions can also be used for Taylor expansions of the kind that are used in the present algorithm, but it is not as efficient in this setting.
Also, since the $B_n^{(m)}$ are derived from the $A_n^{(m)}$ by a convolution, one can do this computation in fewer than the $V^2$ steps of the
straightforward algorithm, by using FFT-based methods.
Greengard and Rokhlin \cite{GR2} report some improvements obtained this way, but
they are only
about
2 or 3 for 2-dimensional problems, and
about
8 for 3-dimensional ones.
Since the zeta function problem is essentially a
1-dimensional
one (with all the poles and points of evaluation on the unit circle), we might expect small improvements from this source.
This might be counteracted to some extent by the fact that the order of expansion $V$ that is used
with the zeta function is higher than in \cite{GR2}, so the overhead might be
smaller,
and noticeable savings might still be obtained.

One aspect of this multipole expansions of Greengard and Rokhlin that would have to be investigated carefully before their algorithm could be used for
the zeta function
computation is its accuracy.
However, based on the results reported so far \cite{GR1,GR2}, that is not likely to be a problem.
\subsection{Computations of low zeros}\label{sc463}
\hspace*{\parindent}
The new algorithm is much more efficient than the implementation of the standard
Riemann-Siegel formula evaluation in \cite{Od2} even around zero number $10^{12}$.
However, this advantage might not hold or be as noticeable around zero $1.5 \times 10^9$,
especially if one were only interested in separating zeros, and not computing them accurately (so that only about 1.2 evaluations of $Z(t)$ would be needed per zero, instead of the
10 or so of the current implementation of the algorithm of \cite{OS}).
Thus if one were interested in extending the numerical verification of the RH beyond the $1.5 \times 10^9$ zeros of \cite{LRW2}, the present implementation
might not help much.
This is due to a large extent to the design of the program, which was aimed at
computing around the $10^{20}$-th zero, and so various parameters were chosen with that goal in mind.
It is likely that the program could be rewritten to be much faster at lower heights, and with more extensive use of $dp$ arithmetic rigorous error analysis could be
performed for it, but this would represent a substantial programming effort.

What we present here is a combination of several techniques that ought to give a simple algorithm for computing $Z(t)$ that ought to be about an order of magnitude
faster than the algorithm of \cite{LRW2}, and for which rigorous error analysis could be performed.
The basic idea is to again compute $Z_1 (t)$ in the standard way, and to
compute $F(t)$ on a uniformly spaced grid of points $T$, $T+ \delta , \dd$, and to use
band-limited function interpolation to then obtain $F(t)$ at intermediate points,
as is explained in Section~\ref{sc43}. The band-limited function interpolation method
errors can be bounded rigorously.
If $k_0 \approx t^{1/4}$,
it would suffice to compute $F(t)$ once every 4 Gram intervals, but to shorten the interpolation
computations and to control the errors better (through having to sum fewer terms in the series in \eqn{eq4315}), it could
be preferable to sample somewhat more frequently.

The evaluation of $F(t)$ in the suggested method
would be performed not by the method of \cite{OS}, but by forming arrays $a_k$ and $b_k$,
$k_0 \le k \le k_1$, with
$$
a_k = 2k^{-1/2} e^{iT \log k} , ~~~~b_k = e^{i \delta \log k} ~.
\eqno{(4.7.3.1)}
$$
$F(T)$ would then be the sum of the $a_k$.
Next, we assign to $a_k$ the value of $a_k \cdot b_k$, and sum the new $a_k$, to obtain $F(T+ \delta )$.
Repeating this operation yields all the $F(T+ j \delta )$.
Since complex $sp$ multiplications and additions are vectorized by the Cray Fortran compiler, this method
would be fast, as was already noted in \cite{Od2}.
(To avoid loss of accuracy in repeated multiplications, it would be advisable to use this method only on short stretches of the grid
$T, T+ \delta , \dd$, so that at $t = T+100 j \delta$, for example, one would
recompute $a_k = 2k^{-1/2} \exp ( it \log k )$ from scratch.)

A further improvement can be obtained by using the Euler product, as was suggested by A. Sch\"{o}nhage
in a slightly different context.
To compute $F(t)$, we do not need to compute all the $2k^{-1/2} \exp ( it \log k)$ for
$k_0 \le k \le k_1$ explicitly.
Instead, we can compute them for all $k$,
$2 \le k \le k_1$, such that $(k,P) = 1$,
where $P = 2 \times 3 \times 5 \times \cdots \times p_h$ is the product of the first $h$
primes (with $h$ small, say $h=4$ or 5).
(This computation would be done by a modification of the method presented above.)
Then, to obtain $F(t)$, we can compute, for each $k$, $1 \le k \le k_1$,
$$
2k^{-1/2} e^{it \log k} \sum_{g \in Q \atop k_0 \le kg \le k_1}
g^{-1/2} e^{it \log g}~,
\eqno{(4.7.3.2)}
$$
where $Q$ is the set of integers all of whose prime factors are $\le p_h$.
Since the sum in (4.7.3.2)
would be the same for many $k$, this operation
would be vectorizable.

The methods presented in this section could also be useful for very accurate computations of high zeros.
If one were to find an extreme example of Lehmer's phenomenon
at large heights,
or
even a suspected counterexample to the RH, where it would be necessary to obtain more accurate values of $Z(t)$
than are given by the present implementation of the algorithm of \cite{OS}, writing an improved version of this algorithm
with a guaranteed error bound would be laborious, and might require a prohibitive amount of time to run.
On the other hand, since values of $Z(t)$ in only a short interval would likely be needed, the method of this section (combined with the suggestion at the end of Section~\ref{sc461} about increased accuracy) might be adequate to resolve any uncertainties.
\subsection*{Acknowledgements}
\hspace*{\parindent}
Since this paper is essentially a continuation of \cite{Od2}, the author would like to thank
P. Barrucand, R. A. Becker,
O. Bohigas, R.~P. Brent, W.~S. Cleveland,
F.~J. Dyson,
P.~X. Gallagher,
E. Grosswald,
D.~A. Hejhal, B. Kleiner, H.~J. Landau, S.~P. Lloyd, C.~L. Mallows,
H.~L. Montgomery, A. Pandey, H.J. J. te Riele, L. Schoenfeld, 
N.~L. Schryer, D. S. Slepian, and D.~T. Winter,
all of whom helped in the preparation of that earlier paper, and many of whom helped
with this one.
Additional help in the
preparation of this paper was
provided by
D.~H. Bailey,
M.~V. Berry, J.~B. Conrey, L. Greengard,
A. Ivi\'{c}
and B.~F. Logan.
D.~A. Hejhal and J. van~de Lune provided exceptionally extensive and useful comments.
A. Sch\"{o}nhage
is due special thanks for the joint work that resulted in the algorithm that
made these computations possible.

The author is grateful to
W.~M. Coughran, P. Glick, R.~H. Knag, and P.~J. Weinberger for help in obtaining time and storage space on computers
and to D.~H. Bailey for providing fast Fourier transform programs.


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\end{thebibliography}


%  FILE: table.tex

\large\normalsize
\renewcommand{\baselinestretch}{1}
\begin{list}
{}{\setlength{\leftmargin}{0.8in}\setlength{\labelwidth}{0.8in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~1.1.~]
Several zeros of the Riemann zeta function near zero number $10^{20}$.
All zeros are of the form $1/2 + i \gamma_n$.
\end{list}

$$
\begin{array}{l|c}
\multicolumn{1}{c|}{n} & \gamma_n -15,202,440,115,920,740,000 \\ \hline
10^{20} - 6 & 7267.894628 \\
10^{20} - 5 & 7267.988948 \\
10^{20} - 4 & 7268.077538 \\
10^{20} - 3 & 7268.258252 \\
10^{20} - 2 & 7268.337163 \\
10^{20} - 1 & 7268.563308 \\
10^{20} & 7268.629029 \\
10^{20} + 1 & 7268.828625 \\
10^{20} + 2 & 7268.972156 \\
10^{20} + 3 & 7269.122460 \\
10^{20} + 4 & 7269.241484 \\
10^{20} + 5 & 7269.313890 \\
\end{array}
$$

\vspace{1in}
\begin{center}
Table~1.2.~~Large computed sets of zeros of the Riemann zeta function. \\
$$
\begin{array}{c|r|l|c}
& & \multicolumn{1}{c|}{\mbox{index of first}} & \multicolumn{1}{c}{\mbox{approximate height}} \\
N & \multicolumn{1}{c|}{\mbox{number of zeros}} & \multicolumn{1}{c|}{\mbox{zero in set}} & \multicolumn{1}{c}{\mbox{of zero no. $N$}} \\ \hline
10^6 & 1,000,1052~~ & N+1 & 6.003 \times 10^5 \\
10^{12} & 1,592,196~~ & N- 6,032 & 2.677 \times 10^{11} \\
10^{14} & 1,685,452~~ & N- 736 & 2.251 \times 10^{13} \\
10^{16} & 16,480,973~~ & N-5,946 & 1.941 \times 10^{15} \\
10^{18} & 16,671,047~~ & N-8,839 & 1.706 \times 10^{17} \\
10^{19} & 16,749,725~~ & N-13,607 & 1.608 \times 10^{18} \\
10^{20} & 175,587,726~~ & N-30,769,710 & 1.520 \times 10^{19} \\
2 \times 10^{20} & 101,305,325~~ & N-633,984 & 2.991 \times 10^{19} \\
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table~2.4.1.~~Moments of $\delta_n -1$. \\

$$
\begin{array}{c|c|c|c|c|c|c|c|c}
k & N=1 & N=10^6 & N= 10^{12} & N=10^{16} & N=10^{18} & N=10^{20} & N=2 \times 10^{20} & \mbox{GUE} \\ \hline
2 & 0.161 & 0.167 & 0.176 & 0.177 & 0.178 & 0.178 & 0.178~~ & 0.180 \\
3 & 0.031 & 0.032 & 0.035 & 0.036 & 0.036 & 0.037 & 0.037~~ & 0.038 \\
4 & 0.081 & 0.088 & 0.096 & 0.098 & 0.098 & 0.099 & 0.099~~ & 0.101 \\
5 & 0.046 & 0.052 & 0.059 & 0.061 & 0.062 & 0.062 & 0.062~~ & 0.066 \\
6 & 0.075 & 0.087 & 0.100 & 0.103 & 0.105 & 0.106 & 0.106~~ & 0.111 \\
7 & 0.072 & 0.089 & 0.109 & 0.113 & 0.115 & 0.116 & 0.116~~ & 0.124 \\
8 & 0.103 & 0.136 & 0.171 & 0.178 & 0.180 & 0.183 & 0.183~~ & 0.197 \\
9 & 0.126 & 0.182 & 0.246 & 0.258 & 0.261 & 0.266 & 0.266~~ & 0.290 \\
10 & 0.181 & 0.283 & 0.408 & 0.431 & 0.434 & 0.444 & 0.444~~ & 0.488
\end{array}
$$
\end{center}

\vspace{1in}
\begin{center}
Table~2.4.2.~~Moments of $\delta_n + \delta_{n+1} - 2$. \\

$$
\begin{array}{c|c|c|c|c|c|c|c|c}
k & N=1 & N=10^6 & N= 10^{12} & N=10^{16} & N=10^{18} & N=10^{20} & N= 2 \times 10^{20} & \mbox{GUE} \\ \hline
2 & 0.207 & 0.218 & 0.236 & 0.241 & 0.242 & 0.243 & 0.243~~ & 0.249 \\        
3 & 0.028 & 0.029 & 0.027 & 0.027 & 0.027 & 0.028 & 0.028~~ & 0.030 \\        
4 & 0.123 & 0.143 & 0.167 & 0.173 & 0.175 & 0.176 & 0.177~~ & 0.185 \\        
5 & 0.047 & 0.057 & 0.062 & 0.064 & 0.065 & 0.066 & 0.066~~ & 0.073 \\        
6 & 0.119 & 0.158 & 0.204 & 0.214 & 0.218 & 0.220 & 0.220~~ & 0.237 \\        
7 & 0.078 & 0.113 & 0.151 & 0.159 & 0.162 & 0.164 & 0.164~~ & 0.185 \\        
8 & 0.155 & 0.246 & 0.370 & 0.393 & 0.401 & 0.406 & 0.407~~ & 0.451 \\        
9 & 0.142 & 0.250 & 0.423 & 0.453 & 0.465 & 0.470 & 0.471~~ & 0.544 \\        
10 & 0.252 & 0.482 & 0.909 & 0.985 & 1.016 & 1.025 & 1.029~~ & 1.178
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table~2.4.3.~~Moments of $\log \delta_n$, $\delta_n^-1$, and $\delta_n^{-2}$. \\

$$
\begin{array}{c|r|r|r|r|r|r|r|r}
\mbox{moments} & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
\mbox{of} & \multicolumn{1}{c|}{N=1} &
\multicolumn{1}{c|}{N=10^6} &
\multicolumn{1}{c|}{N=10^{12}} &
\multicolumn{1}{c|}{N=10^{16}} &
\multicolumn{1}{c|}{N=10^{18}} &
\multicolumn{1}{c|}{N=10^{20}} &
\multicolumn{1}{c|}{N= 2 \times 10^{20}} & \mbox{GUE} \\ \hline
\log \delta_n & -0.0912 & -0.0960 & -0.1013 & -0.1022 & -0.1025 & -0.1027 & -0.1027~~~ & -0.1035 \\
\delta_n^{-1} & 1.2363 & 1.2534 & 1.2700 & 1.2725 & 1.2733 & 1.2737 & 1.2738~~~ & 1.2758 \\
\delta_n^{-2} & 2.2235 & 2.4153 & 2.5277 & 2.5309 & 2.5855 & 2.5475 & 2.5545~~~ & 2.5633
\end{array}
$$
\end{center}

\vspace{1in}
\begin{center}
Table~2.4.4.~~Kolmogorov statistic for $\delta_n$ and $\delta_n + \delta_{n+1}$, for blocks of $10^6$ zeros. \\

$$
\begin{array}{c||c|c||c|c}
~ & \multicolumn{2}{c||}{\delta_n} &
\multicolumn{2}{c}{\delta_n + \delta_{n+1}} \\ \cline{2-5}
~ & D & \mbox{prob.} & D & \mbox{prob.} \\ \hline
N=10^{12}~\mbox{vs. GUE} & 0.00419 & 10^{-15} & 0.00819 & 10^{-58} \\
N=10^{20} (a)~\mbox{vs. GUE} & 0.00180 & 3 \times 10^{-3} & 0.00318 & 3 \times 10^{-9} \\
N=10^{20} (b)~\mbox{vs. GUE} & 0.00152 & 2 \times 10^{-2} & 0.00399 & 3 \times 10^{-14} \\
N=10^{20} (a)~\mbox{vs.~} N=10^{20} (b) & 0.00108 & 0.19 & 0.00119 & 0.12 \\
N=10^{20} (a)~\mbox{vs.~} N=10^{20} (c) & 0.00082 & 0.51 & 0.00123 & 0.10 \\
N=10^{20} (b)~\mbox{vs.~} N-10^{20} (c) & 0.00089 & 0.41 & 0.00096 & 0.32
\end{array}
$$
\end{center}
\clearpage
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.5.1.~]
Moments of scaled values of $S(t)$ computed from two intervals of $10^6$
zeros each near $N=10^{12}$ and $10^{20}$.
\end{list}

$$
\begin{array}{c|c|c|c}
k & N=10^{12} & N=10^{20} & \mbox{normal} \\ \hline
1 & 1.2 \times 10^{-5} & -6.3 \times 10^{-6} & ~~0 \\
2 & 1.0 & 1.0 & ~~1 \\
3 & 3.9 \times 10^{-4} & -4.7 \times 10^{-4} & ~~0 \\
4 & 2.792 & 2.831 & ~~3 \\
5 & 4.8 \times 10^{-3} & -9.1 \times 10^{-3} & ~~0 \\
6 & 12.22 & 12.71 & ~15 \\
7 & 0.050 & -0.140 & ~~0 \\
8 & 70.98 & 76.57 & 105 \\
~ & ~ & ~ & ~ \\
|1| & 0.8058 & 0.8042 & 0.79788 \ldots \\
|3| & 1.3130 & 1.3458 & 1.5957 \ldots \\
|5| & 5.597 & 5.742 & 6.3830 \ldots \\
~ & ~ & ~ & ~ \\
1 \ast & 5.9 \times 10^{-6} & -3.2 \times 10^{-6} & ~ \\
2 \ast & 0.2330808 & 0.2606901 & ~
\end{array}
$$

\vspace{1in}
\begin{center}
Table~2.5.2.~~Average number of sign changes of $S(t)$ per Gram interval. \\

$$
\begin{array}{c|c}
N & S(t)~ \mbox{sign changes} \\ \hline
10^6 & 1.719 \\
10^{12} & 1.600 \\
10^{14} & 1.575 \\
10^{16} & 1.556 \\
10^{18} & 1.538 \\
10^{19} & 1.531 \\
10^{20} & 1.524 \\
2 \times 10^{20} & 1.522 \\
\end{array}
$$
\end{center}
\clearpage
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.5.3.~]
Largest values of $|S(t)|$ in various data sets and fraction of exceptions to
Rosser's rule that had $|S(t)| > 2.3$.
\end{list}

$$
\begin{array}{c|r|c}
~ & ~ & \mbox{fraction of cases} \\
N & \mbox{largest $S(t)$} & \mbox{with $|S(t)| > 2.3$} \\ \hline
10^{12} & 2.1918~~ & \mbox{--} \\
10^{14} & -2.2784~~ & \mbox{--} \\
10^{16} & -2.4639~~ & 0.0123 \\
10^{18} & 2.6121~~ & 0.0175 \\
10^{19} & -2.5698~~ & 0.0162 \\
10^{20} & 2.7916~~ & 0.0240 \\
2 \times 10^{20} & 2.6271~~ & 0.0224 \\
\end{array}
$$

\vspace{1in}
\begin{center}
Table~2.5.4.~~Statistics of $S_1 (t)$. \\

$$
\begin{array}{c|r@{}l|r@{}l}
~ & \multicolumn{2}{c|}{N=10^{12}} & \multicolumn{2}{c}{N=10^{20}} \\ \hline
\mbox{mean of $S_1 (t)^2$} & & 0.0793 & & 0.0793 \\
\mbox{mean of $S_1 (t)^3$} & & 0.0058 & & 0.0058 \\
\mbox{mean of $S_1 (t)^4$} & & 0.0148 & & 0.0148 \\
\max S_1 (t) & & 0.966 & & 0.996 \\
\min S_1 (t) & - & 0.786 & - & 0.768 \\
\mbox{no. sign changes} & & 0.120 & & 0.074
\end{array}
$$
\end{center}
\clearpage
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.6.1.~]
Extremal values of $\delta_n$ and $\delta_n + \delta_{n+1}$, and the probability that the minimum value of $\delta_n$ in the GUE in a sample of the
same size would not exceed the minimal value that was found.
\end{list}

$$
\begin{array}{c|c|c|c|c|l}
~ & ~ & ~ & ~ & ~ & ~\mbox{prob.} \\
N & \min \delta_n & \max \delta_n & \min ( \delta_n + \delta_{n+1} ) & \max ( \delta_n + \delta_{n+1} ) & ~\min \delta_n \\ \hline
10^6 & 0.00545 & 3.3035 & 0.2914 & 4.0683 & ~0.16 \\
10^{12} & 0.00649 & 3.5098 & 0.2952 & 4.5833 & ~0.38 \\
10^{14} & 0.00935 & 3.4716 & 0.2723 & 4.6564 & ~0.78 \\
10^{16} & 0.00454 & 4.1637 & 0.1664 & 4.9921 & ~0.82 \\
10^{18} & 0.00112 & 3.9869 & 0.1680 & 5.0401 & ~0.025 \\
10^{19} & 0.00090 & 3.8089 & 0.1918 & 5.0588 & ~0.013 \\
10^{20} & 0.00197 & 4.0258 & 0.1124 & 5.2125 & ~0.77 \\
2 \times 10^{20} & 0.00121 & 4.0215 & 0.1377 & 5.0859 & ~0.18
\end{array}
$$

\vspace{1in}
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.6.2.~]
Frequencies of large and small $\delta_n$ and $\delta_n + \delta_n$ (number of cases per million zeros) and
the GUE predictions.
\end{list}

$$
\begin{array}{c|c|c|c|c|r}
N & \delta_n \le 0.05 & \delta_n \le 0.1 & \delta_n \ge 2.8 & \delta_n + \delta_{n+1} \le 0.6 & \delta_n + \delta_{n+1}  \ge 4 \\ \hline
10^6 & 121.9 & ~945 & ~87.0 & 126.9 & 8.0~~~~~ \\
10^{12} & 126.2 & 1055 & 157.6 & 331.0 & 94.8~~~~~ \\
10^{14} & 118.7 & 1103 & 156.6 & 329.9 & 97.9~~~~~ \\
10^{16} & 130.9 & 1070 & 164.4 & 341.1 & 107.7~~~~~ \\
10^{18} & 135.3 & 1088 & 169.9 & 356.1 & 108.5~~~~~ \\
10^{19} & 140.5 & 1084 & 170.2 & 362.5 & 114.0~~~~~ \\
10^{20} & 134.7 & 1077 & 174.3 & 363.3 & 111.9~~~~~ \\
2 \times 10^{20} & 135.8 & 1074 & 172.8 & 366.8 & 112.4~~~~~ \\
~ & ~ & ~ & ~ & ~ & ~  \\
\mbox{GUE} & 136.8 & 1088 & 196.8 & 386.3 & 135.7~~~~~
\end{array}
$$
\clearpage
\begin{center}
Table~2.7.1~Autocovariances of the $\delta_n$. \\
\small
$$
\begin{array}{r|r|r|r}
\multicolumn{1}{c|}{k} &
\multicolumn{1}{c|}{N=1} &
\multicolumn{1}{c|}{N=10^{12}} &
\multicolumn{1}{c}{N=10^{20}} \\ \hline
0 & .1607429 & 0.1754737 & 0.1781405 \\
1 & -.0574023 & -0.0576441 & -0.0566976 \\
2 & -.0126083 & -0.0143034 & -0.0143122 \\
3 & -.0065874 & -0.0055030 & -0.0065465 \\
4 & -.0045317 & -0.0026406 & -0.0028474 \\
5 & -.0031454 & -0.0016681 & -0.0019375 \\
6 & -.0011362 & -0.0013422 & -0.0014018 \\
7 & -.0007084 & -0.0009186 & -0.0006824 \\
8 & -.0013904 & -0.0010702 & -0.0006266 \\
9 & .0013483 & -0.0007598 & -0.0005397 \\
10 & .0034456 & -0.0006851 & -0.0004818 \\
11 & .0018714 & -0.0006116 & -0.0002820 \\
12 & -.0002503 & -0.0004058 & -0.0004115 \\
13 & -.0005412 & -0.0006459 & -0.0003212 \\
14 & .0025227 & -0.0005569 & -0.0003363 \\
15 & .0046388 & -0.0007091 & -0.0003671 \\
16 & .0025451 & -0.0001529 & -0.0001061 \\
17 & .0010829 & -0.0000236 & -0.0004597 \\
18 & -.0001093 & 0.0004387 & -0.0000046 \\
19 & -.0057139 & 0.0001141 & -0.0003378 \\
20 & -.0133596 & -0.0000075 & 0.0000028 \\ \hline
9980 &  & 0.0020484 & 0.0018166 \\
9981 &  & -0.0037100 & 0.0012394 \\
9982 &  & -0.0030168 & 0.0003898 \\
9983 &  & 0.0029465 & -0.0015079 \\
9984 &  & 0.0043783 & -0.0019355 \\
9985 &  & -0.0010326 & -0.0012999 \\
9986 &  & -0.0034815 & 0.0001715 \\
9987 &  & 0.0000487 & 0.0014113 \\
9988 &  & -0.0012679 & 0.0021382 \\
9989 &  & -0.0037964 & 0.0004500 \\
9990 &  & 0.0003175 & -0.0005050 \\
9991 &  & 0.0048778 & -0.0014679 \\
9992 &  & 0.0062130 & -0.0018540 \\
9993 &  & 0.0053806 & -0.0002132 \\
9994 &  & 0.0011459 & 0.0014712 \\
9995 &  & -0.0048852 & 0.0013364 \\
9996 &  & -0.0057967 & 0.0010678 \\
9997 &  & -0.0056723 & 0.0001780 \\
9998 &  & -0.0034737 & -0.0014741 \\
9999 &  & 0.0031196 & -0.0020779 \\
10000 &  & 0.0074084 & -0.0014374
\end{array}
$$
\end{center}
\clearpage
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.8.1.~]
Frequency of the Lehmer phenomenon in the $N=10^{19}$, $N=10^{20}$, and $10 = 2 \times 10^{20}$ data sets.
\end{list}

$$
\begin{array}{l|r}
\multicolumn{1}{c|}{x} & \mbox{no. values $< x$} \\ \hline
0.0005 & 6190~~~~~ \\
0.0004 & 4422~~~~~ \\
0.0003 & 2877~~~~~ \\
0.0002 & 1534~~~~~ \\
0.0001 & 573~~~~~ \\
0.00005 & 208~~~~~ \\
0.00002 & 61~~~~~ \\
0.00001 & 24~~~~~ \\
\end{array}
$$
\clearpage
\begin{center}
Table~2.9.1.~Largest values of $| \zeta (1/2 +it)|$ that were found. \\

$$
\begin{array}{c|c}
N & \max |Z(t)| \\ \hline
10^{12} & 176 \\
10^{14} & 246 \\
10^{16} & 460 \\
10^{18} & 376 \\
10^{19} & 448 \\
10^{20} & 641 \\
2 \times 10^{20} & 628
\end{array}
$$
\end{center}

\vspace{1in}
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.9.2.~]
Frequency of large values of $| \zeta (1/2 + it)|$, $N=10^{19}$, $N=10^{20}$, and $N= 2 \times 10^{20}$.
\end{list}

$$
\begin{array}{c|r}
x & \mbox{no. values $> x$} \\ \hline
250 & 1851~~~~~~ \\
300 & 671~~~~~~ \\
350 & 288~~~~~~ \\
400 & 111~~~~~~ \\
450 & 46~~~~~~ \\
500 & 17~~~~~~ \\
\end{array}
$$
\clearpage
\begin{center}
Table~2.10.1.~~Mean values of $| \zeta (1/2 + it)|$. \\

$$
\begin{array}{r|r@{}l|r@{}l|l|r@{}l}
\multicolumn{1}{c|}{\lambda} &
\multicolumn{2}{c|}{r ( \lambda ,H)} &
\multicolumn{2}{c|}{c_1 ( \lambda )} &
\multicolumn{1}{c|}{c_2 ( \lambda )} &
\multicolumn{2}{c}{c( \lambda )} \\ \hline
.1 & 1&.004 & 1&.0042 & ~ & ~ \\
.2 & 1&.034 & 1&.0172 & ~ & ~ \\
.3 & 1&.067 & 1&.0381 & ~ & ~ \\
.4 & 1&.098 & 1&.0640 & ~ & ~ \\
.5 & 1&.123 & 1&.0904 & ~ & ~ \\
.6 & 1&.135 & 1&.1113 & ~ & ~ \\
.7 & 1&.132 & 1&.1195 & ~ & ~ \\
.8 & 1&.107 & 1&.1076 & ~ & ~ \\
.9 & 1&.060 & 1&.0690 & ~ & ~ \\
1.0 & &.989 & 1&.0 & 1.0 & 1&.0 \\
1.1 & &.896 & &.901 & 0.906 & ~ \\
1.2 & &.787 & &.776 & 0.795 & ~ \\
1.3 & &.667 & &.637 & 0.672 & ~ \\
1.4 & &.554 & &.494 & 0.544 & ~ \\
1.5 & &.426 & &.360 & 0.421 & ~ \\
1.6 & &.319 & &.246 & 0.309 & ~ \\
1.7 & &.229 & &.157 & 0.215 & ~ \\
1.8 & &.156 & &.092 & 0.142 & ~ \\
1.9 & &.101 & &.050 & 0.086 & ~ \\
2.0 & &.0624 & &.025 & 0.051 & ~ &.051 \\
2.1 & &.0364 & &.012 & ~ & ~ \\
2.2 & &.0201 & &.0049 & ~ & ~ \\
2.3 & &.0105 & &.0019 & ~ & ~ \\
2.4 & &.00522 & &.00066 & ~ & ~ \\
2.5 & &.00239 & &.00021 & ~ & ~ 
\end{array}
$$
\end{center}

\vspace{.5in}
\begin{center}
Table~2.10.2.~~Negative moments of $| \zeta (1/2 + it)|$. \\

$$
\begin{array}{c|c}
\lambda & \mbox{mean values of $|Z(t)|^{- 2 \lambda}$} \\ \hline
0.1 & 1.06 \\
0.2 & 1.27 \\
0.3 & 1.83 \\
0.4 & 3.77
\end{array}
$$
\end{center}
\clearpage
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.11.1.~]
Moments of the scaled distribution of log $| \zeta (1/2 + it)|$ obtained from $10^6$ random
samples near zero number $N$ and the moments of the normal distribution.
\end{list}

\small
$$
\begin{array}{r|r@{}l|r@{}l|r@{}l|r@{}l|r@{}l|r}
\multicolumn{1}{c|}{k} &
\multicolumn{2}{c|}{N=10^{12}} &
\multicolumn{2}{c|}{N=10^{18} (a)} &
\multicolumn{2}{c|}{N=10^{18} (b)} &
\multicolumn{2}{c|}{N=10^{20} (c)} &
\multicolumn{2}{c|}{N=10^{20} (d)} &
\mbox{normal} \\ \hline
1 & 0&.0 & 0&.0 & 0&.0 & 0&.0 & 0&.0 & 0~~ \\
2 & 1&.0 & 1&.0 & 1&.0 & 1&.0 & 1&.0 & 1~~ \\
3 & -0&.61867 & -0&.54505 & -0&.54199 & -0&.53625 & -0&.55069 & 0~~ \\
4 & 4&.1319 & 3&.9441 & 3&.9491 & 3&.9233 & 3&.9647 & 3~~ \\
5 & -9&.0528 & -7&.8024 & -7&.8610 & -7&.6238 & -7&.8839 & 0~~ \\
6 & 44&.065 & 39&.717 & 40&.360 & 38&.434 & 39&.393 & 15~~ \\
7 & -175&.39 & -159&.45 & -162&.86 & -144&.78 & -148&.77 & 0~~ \\
8 & 900&.06 & 930&.19 & 930&.70 & 758&.57 & 765&.54 & 105~~ \\
9 & -4700&.06 & -6065&.28 & -5692&.4 & -4002&.5 & -3934&.7 & 0~~ \\
10 & 27016&.2 & 48430&.0 & 40818&.3 & 24060&.5 & 22722&.9 & 945~~ \\
~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
1 \ast & -0&.0003725 & -0&.0009607 & 0&.00101075 & -0&.00159534 & 0&.00054934 \\
2\ast & 2&.29679 & 2&.52283 & 2&.51805 & 2&.57360 & 2&.51778
\end{array}
$$
\clearpage
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.12.1.~]
Moments of scaled values of $\log | \zeta' (1/2 + it) |$ computed from $10^6$ zeros for $N=10^{20}$.
\end{list}

$$
\begin{array}{c|r@{}l|r}
~ & ~ & ~ & \multicolumn{1}{c}{\mbox{moments of}} \\
\multicolumn{1}{c|}{k} &
\multicolumn{2}{c|}{\mbox{scaled moments of $\log | Z' ( \gamma )|$}} &
\multicolumn{1}{c}{\mbox{normal distribution}} \\ \hline
1 &~~~~~~~0&.0 & 0~~~~~~~~~~ \\
2 & ~~~~~~~1&.0 & 1~~~~~~~~~~ \\
3 & ~~~~~~~-0&.03377 & 0~~~~~~~~~~ \\
4 & ~~~~~~~3&.0182 & 3~~~~~~~~~~ \\
5 & ~~~~~~~-0&.59687 & 0~~~~~~~~~~ \\
6 & ~~~~~~~15&.4522 & 15~~~~~~~~~~ \\
7 & ~~~~~~~-9&.0568 & 0~~~~~~~~~~ \\
8 & ~~~~~~~115&.378 & 105~~~~~~~~~~ \\
9 & ~~~~~~~-144&.031 & 0~~~~~~~~~~ \\
10 & ~~~~~~~1180&.33 & 945~~~~~~~~~~ \\
~ & ~ & ~ & ~~~~~~~~~~~ \\
1\ast & ~~~~~~~3&.34571 & ~~~~~~~~~~~ \\
2\ast & ~~~~~~~12&.3312~~~~~~~~~~
\end{array}
$$

\vspace{.2in}
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.12.2.~]
Moments of $| \zeta' ( 1/2 +i \gamma_n )|$ divided by conjectured main term,
two sets of zeros near zero number $10^{20}$.
\end{list}

$$
\begin{array}{r@{}l|l|l}
\multicolumn{2}{c|}{\lambda} & \multicolumn{1}{c|}{\mbox{first set of $5 \times 10^5$ zeros}} & \multicolumn{1}{c}{\mbox{second set of $5 \times 10^5$ zeros}} \\ \hline
-&5. & ~~~6.08 \times 10^{-22} & ~~~4.68 \times 10^{-20} \\
-&4.5 & ~~~1.29 \times 10^-16 & ~~~5.40 \times 10^{-15} \\
-&4. &~~~4.21 \times 10^-12 & ~~~9.45 \times 10^{-11} \\
-&3.5 & ~~~2.15 \times 10^{-8} & ~~~2.54 \times 10^{-7} \\
-&3. & ~~~1.74 \times 10^{-5} & ~~~1.07 \times 10^{-4} \\
-&2.5 & ~~~2.29 \times 10^{-3} & ~~~7.41 \times 10^{-3} \\
-&2. & ~~~5.29 \times 10^{-2} & ~~~9.51 \times 10^{-2} \\
-&1.5 & ~~~0.275 & ~~~0.322 \\
-&1. & ~~~0.640 & ~~~0.644 \\
-&0.5 & ~~~1.079 & ~~~1.078 \\
&0. & ~~~1.0 & ~~~1.0 \\
&0.5 & ~~~0.436 & ~~~0.436 \\
&1. & ~~~8.17 \times 10^{-2} & ~~~8.17 \times 10^{-2} \\
&1.5 & ~~~5.31 \times 10^{-3} & ~~~5.32 \times 10^{-3} \\
&2. & ~~~9.26 \times 10^{-5} & ~~~9.48 \times 10^{-5} \\
&2.5 & ~~~3.53 \times 10^{-7} & ~~~3.83 \times 10^{-7} \\
&3. & ~~~2.59 \times 10^{-10} & ~~~3.10 \times 10^{-10} \\
&3.5 & ~~~3.38 \times 10^{-14} & ~~~4.65 \times 10^{-14} \\
&4. & ~~~7.50 \times 10^-19 & ~~~1.22 \times 10^{-18} \\
&4.5 & ~~~2.73 \times 10^{-2}4 & ~~~5.34 \times 10^{-2}4 \\
&5. & ~~~1.59 \times 10^{-30} & ~~~3.79 \times 10^{-30} \\
\end{array}
$$
\clearpage
\begin{center}
Table~2.13.1.~~Fractions of Gram blocks of various lengths. \\

\small
$$
\begin{array}{c|c|c|c|c|l|c|c|c}
N & k=1 & k=2 & k=3 & k=4 & \multicolumn{1}{c|}{k=5} & k=6 & k=7 & \ge 8 \\ \hline
1 & 0.8449 & 0.1249 & 0.0258 & 0.0041 & 3.1 \times 10^{-4} & 1.7 \times 10^{-5} & 6.4 \times 10^{-7} & 0 \\
1.4 \times 10^8 & 0.8325 & 0.1289 & 0.0305 & 0.0069 & 1.03 \times 10^{-3} & 8.2 \times 10^{-5} & 6.0 \times 10^{-6} & 2.8 \times 10^{-7} \\
10^{12} & 0.8178 & 0.1326 & 0.0356 & 0.0106 & 2.8 \times 10^{-3} & 5.4 \times 10^{-4} & 6.7 \times 10^{-5} & 9.4 \times 10^{-6} \\
10^{14} & 0.8099 & 0.1347 & 0.0380 & 0.0122 & 3.9 \times 10^{-3} & 1.1 \times 10^{-3} & 2.1 \times 10^{-4} & 4.3 \times 10^{-5} \\
10^{16} & 0.8045 & 0.1357 & 0.0393 & 0.0135 & 4.8 \times 10^{-3} & 1.6 \times 10^{-3} & 4.5 \times 10^{-4} & 1.1 \times 10^{-4} \\
10^{18} & 0.7998 & 0.1364 & 0.0407 & 0.0147 & 5.5 \times 10^{-3} & 2.1 \times 10^{-3} & 6.9 \times 10^{-4} & 2.5 \times 10^{-4} \\
10^{19} & 0.7977 & 0.1368 & 0.0412 & 0.0150 & 5.9 \times 10^{-3} & 2.3 \times 10^{-3} & 8.2 \times 10^{-4} & 3.3 \times 10^{-4} \\
10^{20} & 0.7957 & 0.1371 & 0.0417 & 0.0155 & 6.2 \times 10^{-3} & 2.5 \times 10^{-3} & 9.3 \times 10^{-4} & 4.3 \times 10^{-4} \\
2 \times 10^{20} & 0.7952 & 0.1372 & 0.0418 & 0.0156 & 6.2 \times 10^{-3} & 2.5 \times 10^{-3} & 9.7 \times 10^{-4} & 4.6 \times 10^{-4} \\
\end{array}
$$
\end{center}

\vspace{1in}
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~2.13.2.~]
Fraction of Gram blocks of given length $k$ that have exactly $k$ zeros and contain a Gram interval
with 3 zeros.
\end{list}

$$
\begin{array}{c|c|c|c|c}
N & k=3 & k=4 & k=5 & k=6 \\ \hline
1 & 0.0511 & 0.0799 & 0.1737 & 0.5448 \\
10^{12} & 0.0449 & 0.0541 & 0.0776 & 0.1032 \\
10^{20} & 0.0356 & 0.0413 & 0.0447 & 0.0392 \\
\end{array}
$$
\clearpage
\begin{center}
Table~2.13.3.~~Fractions of Gram intervals that contain $m$ zeros,
and the GUE prediction. \\

$$
\begin{array}{c|c|c|c|c|c}
N & m=0 & m=1 & m=2 & m=3 & m=4 \\ \hline
1 & 0.13197 & 0.73772 & 0.12864 & 0.00167 & 10^{-8} \\
1.4 \times 10^9 & 0.13965 & 0.72254 & 0.13598 & 0.00183 & 3 \times 10^{-8} \\
10^{12} & 0.14787 & 0.70625 & 0.14388 & 0.00200 & \mbox{--} \\
10^{20} & 0.15748 & 0.68709 & 0.15339 & 0.00204 & \mbox{--} \\
\mbox{GUE} & 0.17022 & 0.66143 & 0.16649 & 0.00186 & 4 \times 10^{-7} 
\end{array}
$$
\end{center}

\vspace{1in}
\begin{center}
Table~2.13.4.~~Averages of $Z(g_n )$ and related functions. \\

$$
\begin{array}{c|c|c}
\mbox{average of} & N=10^{12} & N=10^{20} \\ \hline
Z(g_n ) & 1.12 \times 10^{-2} & -8.801 \times 10^{-4} \\
~ & ~ & ~ \\ [-.1in]
|Z(g_n ) | & 2.6213 & 2.952707053204 \\
~ & ~ & ~ \\ [-.1in]
(-1)^nZ(g_n ) & 2.0000 & 2.0000 \\
~ & ~ & ~ \\ [-.1in]
Z(g_n )^2 & 27.65 & 45.47 \\
~ & ~ & ~ \\ [-.1in]
(-1)^n Z(g_n )^2 & 0.1415 & -0.1945 \\
~ & ~ & ~ \\ [-.1in]
Z(g_n )^3 & 5.539 & 104.98 \\
~ & ~ & ~ \\ [-.1in]
|Z(g_n )^3 | & 749.8 & 2240.4 \\
~ & ~ & ~ \\ [-.1in]
(-1)^n Z(g_n )^3 & 692.7 & 1919.1 \\
~ & ~ & ~ \\ [-.1in]
Z(g_n )^4 & 37645 & 238921 \\
~ & ~ & ~ \\ [-.1in]
(-1)^n Z(g_n )^4 & 110.3 & 31305 \\
~ & ~ & ~ \\ [-.1in]
Z(g_n )^6 & 2.821 \times 10^8 & 1.062 \times 10^{10} \\
~ & ~ & ~ \\ [-.1in]
(-1)^6 Z(g_n )^6 & 1.175 \times 10^6 & 5.803 \times 10^9 \\
~ & ~ & ~ \\ [-.1in]
Z(g_n ) Z(g_{n+1} ) & -3.1387 & -3.1606 \\
~ & ~ & ~ \\ [-.1in]
|Z(g_n ) Z(g_{n+1} ) | & 13.028 & 22.122 \\
~ & ~ & ~ \\ [-.1in]
(-1)^n Z(g_n )Z(g_{n+1} ) & -7.73 \times 10^{-3} & 0.282 \\
~ & ~ & ~ \\ [-.1in]
Z(g_n )^2 Z(g_{n+1} )^2 & 6068 & 46070 \\
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table~2.14.1.~~Number of exceptions to Rosser's rule. \\

$$
\begin{array}{c|r|r@{}l}
~&~&\multicolumn{2}{c}{\mbox{exceptions per}} \\
N & \multicolumn{1}{c|}{\mbox{no. exceptions}} & \multicolumn{2}{c}{\mbox{million zeros}} \\ \hline
10^{12} & 38~~~~~~ & 23&.9 \\
10^{14} & 87~~~~~~ & 51&.6 \\
10^{16} & 1539~~~~~~ & 93&.4 \\
10^{18} & 2453~~~~~~ & 147&.1 \\
10^{19} & 2780~~~~~~ & 166&.0 \\
10^{20} & 34570~~~~~~ & 196&.9 \\
2 \times 10^{20} & 21061~~~~~~ & 207&.9
\end{array}
$$
\end{center}

\vspace{.5in}
\begin{center}
Table~2.14.2.~~Relative frequencies of the most frequent types of exceptions to Rosser's rule. \\

$$
\begin{array}{l|c|c}
& \mbox{first} & \\
\multicolumn{1}{c|}{\mbox{type}} & 1.5 \times 10^9 ~\mbox{zeros} & N=10^{12} , \dd ,10^{20} , 2 \times 10^{20} \\ \hline
2L22 & 0.0363 & 0.1040 \\
2R22 & 0.0373 & 0.1022 \\
2L3 & 0.4501 & 0.0973 \\
2R3 & 0.4386 & 0.0965 \\
3R22 & \mbox{--} & 0.0553 \\
3L22 & \mbox{--} & 0.0553 \\
3R3 & 0.0101 & 0.0529 \\
3L3 & 0.0151 & 0.0517 \\
2L212 & \mbox{--} & 0.0502 \\
2R212 & \mbox{--} & 0.0497 \\
3L212 & \mbox{--} & 0.0260 \\
3R212 & \mbox{--} & 0.0248 \\
4R3 & \mbox{--} & 0.0217 \\
4L3 & \mbox{--} & 0.0214 \\
4L22 & \mbox{--} & 0.0207 \\
4R22 & \mbox{--} & 0.0201 \\
2R2112 & \mbox{--} & 0.0197 \\
2L2112 & \mbox{--} & 0.0187 \\
4L212 & \mbox{--} & 0.0081 \\
3L2112 & \mbox{--} & 0.0079 \\
& ~ & \\
\multicolumn{1}{c|}{\mbox{total}} & 0.9892 & 0.8961
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table 3.2.1.~~Special points for the zeta function. \\

\small
$$
\begin{array}{c|r|c|c|r}
~ & \multicolumn{1}{c|}{\mbox{index of}} &
\mbox{approx. index} & ~ & \\
\mbox{set} & \multicolumn{1}{c|}{\mbox{first zero}} & \mbox{of first zero} & \mbox{no. zeros} &
\multicolumn{1}{c}{\mbox{special point}} \\ \hline
A & 1789820229889768 & 1.8 \times 10^{15} & 213298 & 366350755915100.830671 \\
B & 3225901860089967 & 3.2 \times 10^{15} & 202337 & 648244850785931.253497 \\
C & 4817290207847018 & 4.8 \times 10^{15} & 224580 & 956149582979864.127715 \\
D & 5097943069948350 & 5.1 \times 10^{15} & 204441 & 1010102804832220.857487 \\
E & 6901069159073074 & 6.9 \times 10^{15} & 206276 & 1354828108521396.144683 \\
F & 18950008168234690 & 1.9 \times 10^{16} & 220040 & 3609764047662162.288453 \\
G & 22460777057881112 & 2.2 \times 10^{16} & 221960 & 4257232978148261.305478 \\
H & 42024941452698132 & 4.2 \times 10^{16} & 230978 & 7821904288693735.919567 \\
I & 51214985107007070 & 5.1 \times 10^{16} & 238512 & 9478467782100661.935759 \\
J & 71764726511399980 & 7.2 \times 10^{16} & 221752 & 13154657441819662.863688 \\
K & 76038726777613110 & 7.6 \times 10^{16} & 242968 & 13915273262098117.070642 \\
L & 76935378855702384 & 7.7 \times 10^{16} & 238556 & 14074693071712087.957658 \\
M & 153808369585296620 & 1.5 \times 10^{17} & 228170 & 27596944669957270.886813 \\
N & 233803646149078564 & 2.3 \times 10^{17} & 242576 & 41467826318647943.357194 \\
O & 253172315703241351 & 2.5 \times 10^{17} & 234879 & 44805187485720884.423354 \\
P & 473670769727688896 & 4.7 \times 10^{17} & 254092 & 82413269794748757.568756 \\
Q & 1250710180558723404 & 1.3 \times 10^{18} & 246054 & 212059301707021086.999247 \\
R & 4710265558902545324 & 4.7 \times 10^{18} & 254632 & 771729629469964785.437895 \\
S & 4795416924536726612 & 4.8 \times 10^{18} & 250812 & 785323253967853754.707393 \\
T & 17623088585596705508 & 1.8 \times 10^{19} & 262932 & 2793650241983592679.318477 \\
U & 32220179491036385680 & 3.2 \times 10^{19} & 263299 & 5032868769288289111.005891 \\
V & 35200636070992171652 & 3.5 \times 10^{19} & 265396 & 5486648117377526447.759269 \\
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table~3.2.2.~~Zeta function at special points. \\

$$
\begin{array}{c|r|r|c|r}
\mbox{set} & \multicolumn{1}{c|}{Z(t)} & \mbox{large $S(t)$} & \max \delta_n & \mbox{zero pattern} \\ \hline
A & -396.2 & -2.4235 & 4.5200 & 22000022 \\
B & 459.7 & -2.3202 & 3.7948 & 301000122 \\
C & -663.5 & -2.6410 & 5.1454 & 220000212 \\
D & 598.6 & 2.7575 & 4.9347 & 21120000212 \\
E & 571.0 & 2.3145 & 3.8490 & 22010002112 \\
F & -523.7 & -2.4394 & 3.9353 & 301000212 \\
G & 843.9 & 2.7022 & 5.0612 & 2210000212 \\
H & -555.7 & 2.3022 & 4.3147 & 2120000212 \\
I & 581.6 & -2.1748 & 4.2731 & 2120000212 \\
J & -720.0 & 2.3142 & 4.4956 & 2120000212 \\
K & -767.4 & 2.6654 & 4.8923 & 2120000122 \\
L & -780.0 & 2.6238 & 4.8025 & 2120000212 \\
M & -831.3 & -2.4475 & 4.4574 & 220000122 \\
N & 724.0 & -2.7654 & 4.5515 & 2111000122 \\
O & -874.6 & -2.6160 & 4.7116 & 2120000122 \\
P & -918.8 & 2.2410 & 4.4529 & 3110000212 \\
Q & -971.3 & -2.6178 & 4.6669 & 221000012112 \\
R & 754.7 & 2.1360 & 3.8989 & 22100010212 \\
S & -1065.2 & -2.5178 & 4.4694 & 2120000122 \\
T & 1036.7 & 2.8747 & 4.8433 & 3110001022 \\
U & 1580.6 & -2.4862 & 4.5683 & 220200001212 \\
V & -1329.5 & 2.8314 & 4.3214 & 221100002112
\end{array}
$$
\end{center}
\clearpage
\begin{center}
\begin{list}
{}{\setlength{\leftmargin}{1in}\setlength{\labelwidth}{1in}
\setlength{\labelsep}{.1in}\hfill}
\item[Table~4.3.1.~]
Profile of rational function evaluation algorithm: computations for
$N=10^{20}$, $R=2^{17}$.
\end{list}

$$
\begin{array}{l|r@{}l}
\multicolumn{1}{c|}{\mbox{step}} & \multicolumn{2}{c}{\mbox{time}} \\ \hline
\mbox{evaluate Taylor series coefficients} & 49&.1\% \\
\mbox{Taylor series expansions, $q \ge 4$} & 6&.5\% \\
\mbox{$q=3$ terms} & 12&.2\% \\
\mbox{$q=2$ terms} & 4&.9\% \\
\mbox{$q=1$ terms} & 7&.4\% \\
\mbox{$q=0$ terms} & 3&.8\% \\
\mbox{evaluate $dp \log (k)$} & 9&.6\% \\
\mbox{compute $a_k ,b_k$, etc.} & 6&.5\%
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table~4.5.1.~~Running times (in minutes) of the main rational function evaluation program. \\

$$
\begin{array}{c|c|c|r|c|l|r}
& & & & \mbox{approx.} & & \\
\mbox{set} & R & k_0 & \multicolumn{1}{c|}{k_1} & T & \multicolumn{1}{c|}{\delta} & \mbox{time} \\ \hline
A & 2^{17} & 500 & 7,635,871 & 3.7 \times 10^{14} & 0.323 & 24 \\
d & 2^{23} & 100 & 17,577,894 & 1.9 \times 10^{15} & 0.37 & 438 \\
H & 2^{17} & 500 & 35,283,065 & 7.8 \times 10^{15} & 0.319 & 86 \\
P & 2^{17} & 500 & 114,527,198 & 8.2 \times 10^{16} & 0.3287 & 261 \\
g & 2^{17} & 100 & 164,755,715 & 1.7 \times 10^{17} & 0.2 & 380 \\
f & 2^{23} & 200 & 164,755,715 & 1.7 \times 10^{17} & 0.33 & 1115 \\
i & 2^{23} & 450 & 505,829,004 & 1.6 \times 10^{18} & 0.313 & 2133 \\
U & 2^{17} & 500 & 894,989,353 & 5.0 \times 10^{18} & 0.3067 & 1975 \\
k & 2^{23} & 450 & 1,555,488,184 & 1.5 \times 10^{19} & 0.28961 & 5250 \\
n & 2^{24} & 450 & 1,555,488,184 & 1.5 \times 10^{19} & 0.2903 & 6099 \\
p & 2^{25} & 450 & 1,555,488,184 & 1.5 \times 10^{19} & 0.2901 & 7537 \\
r & 2^{24} & 450 & 2,181,996,752 & 3.0 \times 10^{19} & 0.29025 & 7998
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table~4.6.1.~~Large sets of zeros, showing duplication of computed values. \\

$$
\begin{array}{c|c|r}
 & \mbox{index of first} &  \\
\mbox{set} & \mbox{zero in set} & \mbox{number of zeros} \\ \hline
a & 10^{12} +1 & 101,053~~~ \\
b & 10^{12} - 6,032 & 1,592,196~~~ \\
c & 10^{16} - 4,930 & 1,584,442~~~ \\
d & 10^{16} - 5,946 & 16,480,973~~~ \\
e & 10^{18} - 8,394 & 1,419,501~~~ \\
f & 10^{18} - 8,839 & 16,671,047~~~ \\
g & 10^{18} + 12,333,574 & 157,608~~~ \\
h & 10^{18} + 12,345,608 & 140,684~~~ \\
i & 10^{19} - 13,607 & 16,749,725~~~ \\
j & 10^{19} - 45,597 & 135,161~~~ \\
k & 10^{20} - 30,769,710 & 16,366,702~~~ \\
l & 10^{20} - 15,409,240 & 16,341,831~~~ \\
m & 10^{20} - 48,778 & 16,388,741~~~ \\
n & 10^{20} + 15,311,688 & 32,811,834~~~ \\
o & 10^{20} - 48,867 & 132,188~~~ \\
p & 10^{20} + 47,110,546 & 65,578,910~~~ \\
q & 10^{20} + 111,678,401 & 33,139,615~~~ \\
r & 2 \times 10^{20} - 633,984 & 33,330,777~~~ \\
s & 2 \times 10^{20} + 31,673,368 & 33,199,868~~~ \\
t & 2 \times 10^{20} + 63,843,862 & 36,827,479~~~
\end{array}
$$
\end{center}
\clearpage
\begin{center}
Table~4.6.2.~~Comparison of values for zeros obtained in different computations. \\

$$
\begin{array}{c|c|c}
\mbox{sets of zeros} & \mbox{max. difference} & \mbox{rms difference} \\ \hline
\mbox{$a$ vs. $b$} & 2.5 \times 10^{-9} & 3.7 \times 10^{-11} \\
\mbox{$c$ vs. $d$} & 5.7 \times 10^{-8} & 3.4 \times 10^{-10} \\
\mbox{$e$ vs. $f$} & 4.2 \times 10^{-8} & 3.3 \times 10^{-10} \\
\mbox{$f$ vs. $g$} & 2.2 \times 10^{-8} & 2.3 \times 10^{-10} \\
\mbox{$g$ vs. $h$} & 3.5 \times 10^{-8} & 1.6 \times 10^{-10} \\
\mbox{$i$ vs. $j$} & 2.6 \times 10^{-8} & 7.5 \times 10^{-10} \\
\mbox{$k$ vs. $l$} & 7.3 \times 10^{-7} & 3.8 \times 10^{-9} \\
\mbox{$l$ vs. $m$} & 7.6 \times 10^{-7} & 5.9 \times 10^{-9} \\
\mbox{$m$ vs. $n$} & 5.8 \times 10^{-7} & 3.9 \times 10^{-9} \\
\mbox{$m$ vs. $o$} & 2.7 \times 10^{-7} & 4.2 \times 10^{-9} \\
\mbox{$n$ vs. $p$} & 5.7 \times 10^{-7} & 4.4 \times 10^{-9} \\
\mbox{$p$ vs. $q$} & 5.2 \times 10^{-7} & 4.1 \times 10^{-9} \\
\mbox{$r$ vs. $s$} & 1.1 \times 10^{-6} & 9.5 \times 10^{-9} \\
\mbox{$s$ vs. $t$} & 4.7 \times 10^{-7} & 7.5 \times 10^{-9}
\end{array}
$$
\end{center}

%  FILE: figure.tex

\begin{center}
\section*{Figure Captions}
\end{center}
\begin{list}
{}{\setlength{\leftmargin}{1.1in}\setlength{\labelwidth}{.9in}
\setlength{\labelsep}{.1in}\hfill}
\item[Figure~2.1.1.]
$Z(t)$ near zero number $10^{20}$.
The horizontal axis extends from Gram point number $10^{20} -8$ to Gram point
number $10^{20} +4$.
\item[Figure~2.1.2.]
$S(t)$ near zero number $10^{20}$.
The range
of $t$ is the same as in Fig.~2.1.1, and the jumps by 1 occur at zeros of the zeta function numbered $10^{20} -6$ to $10^{20} +5$.
\item[Figure~2.1.3.]
$Z(t)$ near zero number $10^{20}$.
The horizontal axis extends from Gram point number $10^{20} - 50$ to Gram point
number $10^{20} + 50$.
\item[Figure~2.4.1.]
Pair correlation of zeros of the zeta function.
Solid line:
GUE prediction.
Scatterplot: empirical data based on $8 \times 10^6$ zeros near zero number $10^{20}$.
\item[Figure~2.4.2.]
Pair correlation of zeros of the zeta function.
Solid line:
GUE prediction.
Scatterplot: empirical data based on
$10^6$ zeros near zero number $10^{12}$.
\item[Figure~2.4.3.]
Pair correlation of zeros of the zeta function.
Solid line:
GUE prediction.
Scatterplot: empirical data based on
$8 \times 10^6$ zeros near zero number $10^{20}$.
Scatterplot smoothed.
\item[Figure~2.4.4.]
Probability density of the normalized spacings $\delta_n$.
Solid line:
GUE prediction.
Scatterplot:
empirical data based on 1,592,196 zeros near zero number $10^{12}$.
\item[Figure~2.4.5.]
Probability density of the normalized spacings $\delta_n$.
Solid line:
GUE prediction.
Scatterplot:
empirical data based on
78,893,234
zeros near zero number $10^{20}$.
\item[Figure~2.4.6.]
Probability density of the normalized spacings $\delta_n + \delta_{n+1}$.
Solid line:
GUE prediction.
Scatterplot:
empirical data based on 1,592,196 zeros near zero number $10^{12}$.
\item[Figure~2.4.7.]
Probability density of the normalized spacings $\delta_n + \delta_{n+1}$.
Solid line:
GUE prediction.
Scatterplot:
empirical data based on
78,893,234
zeros near zero number $10^{20}$.
\item[Figure~2.5.1.]
Comparison of the scaled distribution of $S(t)$ for $N=10^{20}$ to the
asymptotic normal distribution.
\item[Figure~2.6.1.]
Initial segment of the quantile-quantile plot of the normalized
spacings $\delta_n$ against the GUE prediction.
Data based on $10^6$ consecutive values of $n$,
starting with $n=10^{20} - 42, 778$.
Straight line $y=x$ drawn to facilitate comparisons.
\item[Figure~2.6.2.]
Initial segment of the quantile-quantile plot of the normalized
spacings $\delta_n$ against the GUE prediction.
Data based on $10^6$ consecutive values of $n$,
starting with
$n= 10^{20} + 15, 316, 087$.
\item[Figure~2.6.3.]
Initial segment of the quantile-quantile plot of the normalized
spacings
$\delta_n$
against the GUE prediction.
Data based on
112,314,003
values of $n$ from $N=10^18$, $10^{19}$, and $10^{20}$ data sets.
\item[Figure~2.6.4.]
Initial segment of the quantile-quantile plot of the normalized
spacings
$\delta_n + \delta_n+1$
against the GUE prediction.
Data based on $10^6$ consecutive values of $n$,
starting with
$n=10^{12} - 6, 032$.
\item[Figure~2.6.5.]
Initial segment of the quantile-quantile plot of the normalized
spacings
$\delta_n + \delta_n+1$
against the GUE prediction.
Data based on $10^6$ consecutive values of $n$,
starting with
$n=10^{20} - 42, 778$.
\item[Figure~2.6.6.]
Find segment of the quantile-quantile plot of the normalized
spacings
$\delta_n$
against the GUE prediction.
Data based on $10^6$ consecutive values of $n$,
starting with
$n=10^{12} - 6, 032$.
\item[Figure~2.6.7.]
Final segment
of the quantile-quantile plot of the normalized
spacings
$\delta_n$
against the GUE prediction.
Data based on $10^6$ consecutive values of $n$,
starting with
$n=10^{20} - 42, 778$.
\item[Figure~2.7.1.]
Graph of $2 \log  | \sum  \exp (i \gamma_n y ) | $,
where $n$ runs over $10^{20} +1 \le n \le 10^{20} + 40,000$,
and values $< 0$ and $> 16$ are deleted.
\item[Figure~2.7.2.]
Graph of $2 \log  | \sum \exp (i \gamma_n y ) | $,
where $n$ runs over $10^{20} +1 \le n \le 10^{20} + 40,000$,
and 
values
$< 0$ are deleted.
\item[Figure~2.7.3.]
Graph of $2 \log  | \sum \exp (i \gamma_n y ) | $,
where $n$ now runs over $10^{20} + 1  \le n \le 10^{20} + 400,000$,
and values $< 0$ are deleted.
\item[Figure~2.7.4.]
Variance of the number of zeros in an interval of length $L$
for the GUE (dashed line), for $5 \times 10^5$ zeros near zero number $10^{20}$
(scatterplot), and Berry's prediction (solid line).
\item[Figure~2.7.5.]
Variance of the number of zeros in an interval of length $L$ based on $5 \times 10^5$ zeros near zero number $10^{20}$.
\item[Figure~2.7.6.]
Variance of the number of zeros in an interval of length $L$ based on $5 \times 10^5$ zeros near zero number $10^{20}$.
\item[Figure~2.8.1.]
Neighborhood of an example of Lehmer's phenomenon.
Graph of $Z(t)$ between Gram points $n-6$ and $n+6$, where
$n=10^{18} + 12, 376, 778$.
The point between Gram points $n+1$ and $n+2$ where $Z(t)$ is seemingly tangent to the zero line represents
2 zeros with $\delta_{n+2} = 0.0011$ and the minimum of $Z(t)$ between those zeros equal to $-5 \times 10^{-7}$.
For a smaller scale picture of this phenomenon, see Fig.~4.6.1.
The other point of near tangency, near Gram point $n-3$, has minimum of $Z(t)$ of
$-0.0094$.
\item[Figure~2.11.1.]
Comparison of the
distribution of $\log  | \zeta ( 1/2 + it )| $ over two ranges of $10^6$ zeros each near
zeros number $10^{12}$ and $10^{20}$ to that of the normal
distribution.
\item[Figure~2.11.2.]
For each $k$, plots the logarithm of the fraction of time that
$| \zeta ( 1/2 + it )| \in [k-1, k)$.
Data obtained from 3 intervals covering $2.8 \times 10^6$ zeros near zero number $10^{20}$.
\item[Figure~2.12.1.]
Scaled distribution of $10^6$ values of $log  | \zeta' ( 1/2 + i \gamma ) | $ for $N=10^{20}$ vs. the conjectured standard normal distribution.
\item[Figure~2.13.1.]
Distribution modulo~1 of $\gamma_n$ on Gram point scale,
for two sets of $10^6$ zeros each.
Curves derived by smoothing a histogram.
\item[Figure~3.2.1.]
$Z(t)$ near the point where the largest known value of $S(t)$ occurs.
The horizontal axis extends from Gram point number $n = 17, 623, 088, 585, 596, 834, 905$ to Gram point
number $n+30$.
\item[Figure~3.2.2.]
$Z(t)$ near the point where the largest known value of $S(t)$ occurs.
The horizontal axis extends from Gram point number $n+9$ to Gram point
number $n+21$, where
$n = 17, 623, 088, 585, 596, 834, 905$.
The high peak of $Z(t)$ has been cut off.
This is a smaller scale view of the central part of Fig.~3.2.1.
\item[Figure~3.2.3.]
$S(t)$ near the point where its largest known value occurs.
The range of $t$ is the same as in Fig.~3.2.2.
\item[Figure~4.6.1.]
Small neighborhood of an example of Lehmer's phenomenon.
Graph of $Z(t)$ on a segment of the interval between Gram points
$n$ and $n+1$ (corresponding to 0 and 1 on the scale of the figure),
where $n=10^{18} + 12, 376, 799$.
Enlargement of a section of Fig.~2.8.1.
\item[Figure~4.6.2.]
Small scale view of Lehmer's phenomenon.
Enlargement of a section of Fig.~4.6.1.
The three curves represent three different computations of $Z(t)$ on this segment.
\end{list}