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{\Large\bf {The $10^{21}$-st zero of the Riemann zeta function}} \bigskip \\
Andrew Odlyzko \\
AT\&T Shannon Laboratory \\
Florham Park, New Jersey \\
amo@research.att.com \bigskip \\
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November 24, 1998 \\
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This is an extremely brief report on computation of large numbers
of high zeros of the Riemann zeta functions, with pointers to
sources of more detailed information.  There have been 
many calculations that verified the Riemann Hypothesis (RH)
for initial sets of zeros of the zeta function.  The largest
of these was that of van de Lune, te Riele, and
Winter \cite{LRW}.  They checked that the first $1.5\times 10^9$ nontrivial
zeros all lie on the critical line.  
Their computations used about 1500 hours on one of the most powerful
computers in existence at that time.  Since then, better algorithms
have been developed, and much more computing power has become
available.  Therefore it would be easy to extend their verification
of the RH, should someone wish to do so.

Starting in the late 1970s, I carried out a series of computations
that not only verified that nontrivial zeros lie on the critical
line, but in addition obtained accurate values of those zeros.
These calculations 
were designed to check the Montgomery pair-correlation
conjecture \cite{Mon}, as well as further conjectures that
predict that zeros of the zeta function
behave like engenvalues of matrices from the GUE.
Instead of starting from the lowest zeros, these computations
obtained values of blocks of consecutive zeros high up in
the critical strip.  The motivation for jumping high up was to come
closer to observing the true
asymptotic behavior of the zeta function, which is often
approached slowly. 

The first computations, done on a Cray supercomputer using
the standard Riemann-Siegel formula, were described in \cite{Od1}.
The highest zeros they included were around zero \# $10^{12}$.
Those calculations stimulated the invention, jointly with
Arnold Sch\"{o}nhage \cite{Od2,OS}, of an improved
algorithm for computing large sets of zeros.  This
algorithm, with some technical improvements, was implemented
and used to compute several hundred million zeros at large
heights, many near zero \# $10^{20}$, and some near zero \# $2 \times 10^{20}$.
Implementation details and results are described in the
manuscripts \cite{Od3,Od4} that have not been published,
but have circulated widely.

During the last couple of years, the algorithms of \cite{Od3,Od4}
have been ported from Cray supercomputers to Silicon Graphics
workstations, and have been used to compute several billion
high zeros of the zeta function.  Some of those zeros are
near zero \# $10^{21}$, and it has been established (not entirely
rigorously, though, as is explained in \cite{Od3,Od4}) that
zeros number $10^{21}-1$, $10^{21}$, and $10^{21}+1$ are
$$
\begin{array}{l}
144176897509546973538.2256529... \\
144176897509546973538.2912188... \\
144176897509546973538.4980696...
\end{array}
$$
These values and many others can be found at
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$\langle$http://www.research.att.com/$\sim$amo/zeta\_tables/index.html$\rangle$.
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Further computations are continuing, and it is likely that
some zeros near zero \# $10^{22}$ will be computed.
A revision of
\cite{Od3,Od4} that describes them will be prepared and
published in the future.  Results will be available
through my home page,
\begin{center}
$\langle$http://www.research.att.com/$\sim$amo$\rangle$.
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Finally, let me mention that many other computations
of zeros of various zeta and L-functions have been
done.  Many are referenced in \cite{Od5}.
There are also interesting new results in the recent
Ph.D. thesis of Michael Rubinstein \cite{Ru}.


\begin{thebibliography}{MMM}
\bibitem[LRW]{LRW}
J. van de Lune, H. J. J. te Riele, and D. T. Winter, On the zeros
of the Riemann zeta function in the critical strip.~IV., 
{\em Math. Comp. 46} (1986), 667--681.

\bibitem[Mon]{Mon}
H. L. Montgomery, The pair correlation of zeros of the zeta
function, pp. 181--193 in {\em Analytic Number Theory,} H. G. Diamond, ed.,
Proc. Symp. Pure Math. {\em 24}, Amer. Math. Soc., Providence 1973.

\bibitem[Od1]{Od1}
A. M. Odlyzko, On the distribution of spacings between zeros of
the zeta function, {\em Math. Comp. 48} (1987), 273--308.
Available at $\langle$http://www.research.att.com/$\sim$amo/doc/complete.html$\rangle$.


\bibitem[Od2]{Od2}
A. M. Odlyzko, New analytic algorithms in number theory,
pp. 466--475 
in {\em Proc. Intern. Congress Math. 1986}, Amer. Math. Soc. 1987.
Available at $\langle$http://www.research.att.com/$\sim$amo/doc/complete.html$\rangle$.

\bibitem[Od3]{Od3}
A. M. Odlyzko,
The $10^{20}$-th zero of the Riemann zeta function and 70 million of its neighbors,
unpublished manuscript, 1989.  Available at
$\langle$http://www.research.att.com/$\sim$amo/unpublished/index.html$\rangle$.

\bibitem[Od4]{Od4}
A. M. Odlyzko,
The $10^{20}$-th zero of the Riemann zeta function and 175 million of its neighbors,
unpublished manuscript, 1992.  Available at
$\langle$http://www.research.att.com/$\sim$amo/unpublished/index.html$\rangle$.

\bibitem[Od5]{Od5}
A. M. Odlyzko, Analytic computations in number theory,
pp.~451--463 in
{\em Mathematics of Computation 1943-1993: A Half-Century of Computational Mathe
matics},
Gautschi, W., ed.
Amer. Math. Soc.,
Proc. Symp. Appl. Math.
\# 48 (1994).
Available at $\langle$http://www.research.att.com/$\sim$amo/doc/complete.html$\rangle$.


\bibitem[OS]{OS}
A. M. Odlyzko and A. Sch\"{o}nhage, Fast algorithms for
multiple evaluations of the Riemann zeta function, {\em Trans.
Amer. Math. Soc. 309} (1988), 797--809.
Available at $\langle$http://www.research.att.com/$\sim$amo/doc/complete.html$\rangle$.

\bibitem[Ru]{Ru}
M. Rubinstein, Evidence for a spectral interpretation of the zeros
of L-functions, Ph.D. thesis, Math. Dept., Princeton Univ., 1998.

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