\documentstyle[twoside,numinsec]{etnaltex}
\pagestyle{myheadings}
\markboth{G. Csordas, A.M. Odlyzko, W. Smith, and R.S. Varga}
{A New Lehmer pair of zeros and a new lower bound} 
\renewcommand{\theequation}{{\arabic{equation}}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\R}{\rm I\kern-.19emR}
\newcommand{\C}{\rm I\kern-.5emC}
\newcommand{\dis}{\displaystyle}
\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Lemma}{Lemma}[section]
\newtheorem{Definition}{Definition}[section]
\newcommand{\proofend}{$\Box$}
\title{A new Lehmer pair of zeros and a new lower bound for the 
de Bruijn-Newman constant {$\Lambda$}\thanks{Received November 24, 1993.
Accepted for publication December 13, 1993. Communicated by A. Ruttan.
Corrected June 29, 1994. The original manuscript is stored in
vol.1.1993/pp104-111.dir/pp104-111orig.ps}} 
\author{
G. Csordas 
\thanks{Department of Mathematics, University of Hawaii, 
Honolulu, HI 96822, USA.
 (e-mail: {\tt george@kahuna.math.hawaii.edu}).}
\and
A.M. Odlyzko 
\thanks{AT\&T Bell Labs, 2C-355, Murray Hill, NJ 07974, USA.
(e-mail: {\tt amo@research.att.com}).}
\and
W. Smith
\thanks{Department of Mathematics, University of Hawaii, Honolulu, HI 96822, 
USA. (e-mail: {\tt wayne@kahuna.math.hawaii.edu}).}
\and
R. S. Varga
\thanks{Institute for Computational Mathematics, Kent State University, 
Kent, OH  44242 (e-mail: {\tt varga@mcs.kent.edu}).}
}

\pagestyle{myheadings}
\markboth{A Lower Bound for the 
de Bruijn-Newman Constant $\Lambda$}
{G. Csordas et. al.}

\begin{document}     
\etnaheader
\addtocounter{page}{103}
\sloppy
\maketitle

\begin{center}
{\em Dedicated to Wilhelm Niethammer on the occasion of his 60th birthday.}
\end{center}

\begin{abstract}
The de Bruijn-Newman constant $\Lambda$ has been investigated extensively
because the truth of the Riemann Hypothesis is equivalent to
the assertion that $\Lambda \leq 0$.  On the other hand, C. M. Newman
conjectured that $\Lambda \geq 0$.  This paper improves previous
lower bounds by showing that
\[
-5.895 \cdot 10^{-9} < \Lambda.
\]
This is done with the help of a spectacularly close pair of
consecutive zeros of the Riemann zeta function.
\end{abstract}

\begin{keywords}
 Lehmer pairs of zeros, de Bruijn-Newman constant, Riemann Hypothesis.
\end{keywords}

\begin{AMS}
30D10, 30D15, 65E05.
\end{AMS}

\section{Introduction}
It is known (cf.\ Titchmarsh [9, p.\ 255]) that the Riemann 
$\xi$-function can be expressed in the form 
\renewcommand{\theequation}{1.1}
\be
\xi \left(\frac{x}{2}\right) /8 = \dis{\int_{0}^{\infty}} ~ 
\Phi (u) \cos (xu) du ~~~~~ (x \in {\C}),
\ee
where
\renewcommand{\theequation}{1.2}
\be
\Phi (u) := \dis{\sum_{n=1}^{\infty}} ~ \left(2\pi^{2} n^{4} e^{9u} - 
3\pi n^{2} e^{5u}\right) \exp \left( -\pi n^{2} e^{4u}\right) ~~~~~ 
(0 \leq u < \infty),
\ee
and the {\em Riemann Hypothesis is the statement that all zeros of
$\xi$ are real}.
If we define
\renewcommand{\theequation}{1.3}
\be
H_{t} (x) := \dis{\int_{0}^{\infty}} ~ e^{tu^{2}} \Phi(u) \cos (xu) du ~~~~~ 
(t \in {\R}; x \in {\C}),
\ee
then $H_{0}$ and the Riemann $\xi$-function are related through 
\renewcommand{\theequation}{1.4}
\be
H_{0} (x) = \xi \left(\frac{x}{2}\right)/8,
\ee
so that the Riemann Hypothesis is also equivalent to the statement 
that all zeros of $H_{0}$ are real. \\

In 1950, De Bruijn [2] established that \\

\begin{tabular}{rp{5in}}
({\em i}) & $H_{t}$ has only real zeros for $t \geq 1/2$; \\
 & \\
({\em ii}) & if $H_{t}$ has only real zeros for some real $t$, then 
$H_{t'}$ has only real zeros \newline for any $t'\geq t$. \\
& \\
\end{tabular}

\noindent C.M. Newman showed further in [6] that there is a real 
constant $\Lambda$, which satisfies $-\infty < \Lambda \leq 1/2$, such that
\renewcommand{\theequation}{1.5}
\be
 H_{t} {\rm \ has\ only\ real\ zeros\ if\ and\ only\ if\ } t \geq \Lambda.
\ee 
In the literature, this constant $\Lambda$ is now called the 
{\bf de Bruijn-Newman constant}.
The Riemann Hypothesis is equivalent to the conjecture that $\Lambda \leq 0$.
On the other hand, C. M. Newman
conjectured that $\Lambda \geq 0$.  The significance of
Newman's conjecture is that if it is true,
then the Riemann Hypothesis, even if it is true, is only barely so,
as even slight perturbations of the zeta function give rise to
zeros that are not on the critical line.

There has been extensive recent research activity in finding 
lower bounds for $\Lambda$,
and these results have been summarized in Csordas, Smith, and Varga [4]. 
In particular, the 
best lower bound for $\Lambda$ in that paper was 
\renewcommand{\theequation}{1.6}
\be
-4.379 \cdot 10^{-6} < \Lambda.
\ee

It is known (cf.\ Csordas, Norfolk, and Varga [3])
that $H_{t}$, defined in (1.3), is an 
{\em even\/} real entire function of order 1 and maximal type, for each 
real $t$. Thus, from the Hadamard factorization theorem, $H_{t} (x)$ can be 
represented as
\renewcommand{\theequation}{1.7}
\be
H_{t} (x) = H_{t} (0) \dis{\prod_{j=1}^{\infty}} ~ 
\left( 1-\frac{x^{2}}{x_{j}^{2}(t)}\right) ~~~~~ (t \in {\R}; x \in {\C}),
\ee
where from (1.3) and from the fact that $\Phi (u) > 0$ for 
all $u \geq 0$, it follows that $H_{t} (0) > 0$.  It is also known that 
\renewcommand{\theequation}{1.8}
\be
\dis{\sum_{j=1}^{\infty}} ~ |x_{j} (t)|^{-2} < \infty.
\ee
It is convenient to order the zeros of $H_{0}$, 
$\{ x_{j} (0)\}_{j=1}^{\infty}$, in ${\rm Re}~z > 0$ 
according to increasing modulus, and, from the evenness of 
$H_{0}$, we set 
\renewcommand{\theequation}{1.9}
\be
x_{-j} (0) := - x_{j} (0) ~~~~~ (j = 1, 2, \cdots).
\ee

Following Csordas, Smith, and Varga [4], we make the following  \\

\begin{Definition}
With $k$ a positive integer, let $x_{k} (0)$ and $x_{k+1} (0)$ (with 
$0 < x_{k} (0) < x_{k+1} (0)$) be two consecutive simple positive 
zeros of $H_{0}$, and set 
\renewcommand{\theequation}{1.10}
\be
\Delta_{k} := x_{k+1} (0) - x_{k} (0).
\ee
Then, $\{ x_{k} (0); x_{k+1} (0)\}$ is a {\bf Lehmer pair of zeros} of 
$H_{0}$ if 
\renewcommand{\theequation}{1.11}
\be
\Delta_{k}^{2} \cdot g_{k} (0) < 4/5,
\ee
where 
\renewcommand{\theequation}{1.12}
\be
g_{k} (0) := \dis{\sum_{j\neq k, k+1}}\!\!\! ' ~ 
\left\{ \frac{1}{(x_{k}(0)-x_{j}(0))^{2}} + 
\frac{1}{(x_{k+1}(0)-x_{j}(0))^{2}}\right\} ;
\ee
\end{Definition}

\noindent
here (and in what follows), the prime in the above summation 
means that $j \neq 0$, so that the above summation extends over all positive
and negative integers with $j \neq k, k+1, 0$. 


We remark that the convergence of the sum in (1.12) is guaranteed by the 
convergence of the sum $\dis{\sum_{j=1}^{\infty}} ~ |x_{j} (0)|^{-2}$ 
(cf.\ (1.8)).  

With Definition 1.1, we further have from 
Csordas, Smith, and Varga [4] the following result. 

\begin{Theorem}
 Let $\{ x_{k} (0); x_{k+1}(0)\}$  be a Lehmer pair of zeros of $H_{0}$. 
If (cf. (1.12)) $g_{k} (0) \leq 0$, then $\Lambda > 0$. 
If $g_{k} (0) > 0$, set 
\renewcommand{\theequation}{1.13}
\be
\lambda_{k} := \frac{(1-\frac{5}{4} \Delta_{k}^{2} \cdot 
g_{k} (0))^{4/5}-1}{8g_{k} (0)},
\ee
so that $-1/[8g_{k}(0)] < \lambda_{k} < 0$.  Then, the de Bruijn-Newman 
constant $\Lambda$  satisfies
\renewcommand{\theequation}{1.14}
\be
\lambda_{k} \leq \Lambda.
\ee
\end{Theorem}

\section{Application of Theorem 1.1}
For our applications below, let $N(T)$ denote the number of 
zeros of the Riemann zeta function $\zeta (s)$, with $s = \sigma + it$, 
in the rectangle $0 \leq \sigma \leq 1$ and $0 \leq t \leq T$. 
The following result was proved by Backlund [1]. \\

\begin{Theorem}
$N(T)$  satisfies 
\renewcommand{\theequation}{2.1}
\be
N(T) = \frac{T}{2\pi} \log \left( \frac{T}{2\pi}\right) - 
\frac{T}{2\pi} + \frac{7}{8} + e (T),
\ee
where
\renewcommand{\theequation}{2.2}
\be
| e (T)| < 0.137 \log T + 0.443 \log \log T + 4.35 ~~~~~ 
(T \geq 2).
\ee
\end{Theorem}

A straightforward calculation, based on (2.1) and (2.2), gives the
following result, whose proof is given (for completeness) in the Appendix. 

\begin{Lemma}
$N(T)$ satisfies
\renewcommand{\theequation}{2.3}
\be
N(T+1) - N(T) \leq \log T ~~~~~ (T \geq 3\cdot 10^{8}).
\ee
\end{Lemma}

This brings us to 

\begin{Lemma}. 
Suppose $\Lambda < 0$, so that all zeros, $x_{j} := x_{j} (0)$, 
of $H_{0}$ are real (cf.\ (1.5)) and recall that 
$x_{j} = 2\gamma_{j}$, 
where $\frac{1}{2} + i \gamma_{j}$ is the associated zero of $\zeta (s)$. 
Then,
\renewcommand{\theequation}{2.4}
\be
\dis{\sum_{j=m}^{\infty}} ~ \frac{1}{x_{j}^{2}} \leq 
\frac{\log ([\gamma_{m}] -1)+1}{4([\gamma_{m}]-1)} ~~~~~ 
([\gamma_{m}] \geq 3 \cdot 10^{8}), 
\ee
where, for each real $u,~[u]$ denotes the greatest integer
$\leq u$. 
\end{Lemma}

\noindent
{\em Proof:}
We have
\[
\begin{array}{rcl}
\dis{\sum_{j=m}^{\infty}} ~ 
\frac{1}{x_{j}^{2}} & = & \dis{\frac{1}{4}} \dis{\sum_{j=m}^{\infty}} ~ 
\frac{1}{\gamma_{j}^{2}} \leq \frac{1}{4} 
\dis{\sum_{j=[\gamma_{m}]}^{\infty}} ~ 
 ~ \dis{\sum_{j\leq \gamma_{\ell} < j+1}} ~ 
\frac{1}{\gamma_{\ell}^{2}} \\
& & \\
& \leq & \dis{\frac{1}{4}} \dis{\sum_{j = [\gamma_{m}]}^{\infty}} ~ 
\left(\frac{N(j+1)-N(j)}{j^{2}}\right) \leq \frac{1}{4} 
\dis{\sum_{j = [\gamma_{m}]}^{\infty}} ~ 
\frac{\log j}{j^{2}},
\end{array}
\]
the last inequality following from (2.3) of Lemma 2.1. 
But, this last sum is bounded above by
\[
\frac{1}{4} \dis{\int_{[\gamma_{m}]-1}^{\infty}} ~ 
\frac{\log u~du}{u^{2}} = \frac{\log ([\gamma_{m}]-1)+1}{4(
[\gamma_{m}]-1)},
\]
which is the desired result of (2.4). \proofend

In their important numerical study of the zeros of the Riemann $\zeta$-function 
on the critical line, van de Lune, te Riele, and Winter [5] found a 
{\em spectacularly\/} close pair of consecutive simple zeros, namely,
$\frac{1}{2} + i \gamma_{K}$ and $\frac{1}{2} + i \gamma_{K+1}$, 
for which (cf.\ (2.8)) 
\[
\gamma_{K+1} - \gamma_{K} = 0.000~108~569~6 ~~~~~ (K := 1,048,449,114).
\]
Then, $2\gamma_{K}$ and $2\gamma_{K+1}$ are zeros of the function $H_0$,
so that (cf.\ (1.4))
\renewcommand{\theequation}{2.5}
\be
\left\{ \begin{array}{rcl}
x_{K} := x_{K} (0) = 2\gamma_{K} & = 
 & 7.777~177~720~045~702~406 \cdot 10^{8}, {\rm \ and} \\
x_{K+1} := x_{K+1} (0) = 2\gamma_{K+1} & = 
   & 7.777~177~720~047~873~798 \cdot 10^{8},
   \end{array}\right.
\ee
is similarly a spectacularly close pair of consecutive simple positive zeros 
of $H_{0}$. 
The calculations of van de Lune, te Riele, and Winter [5] 
established that the first  $1.5 \cdot 10^{9}$ zeros
are real, but they did not
compute accurate values for them.
% basically found {\em intervals\/} containing zeros of the $\zeta$-function, 
Therefore, we have used a CRAY-YMP and techniques from 
Odlyzko [7] to determine,
to high precision, a large number of zeros of $H_{0}$ 
on either side of the zeros of (2.5), in order to facilitate the estimation 
of $g_{K} (0)$ of (1.12). As we shall see below, only a surprisingly 
small number of these nearby zeros is actually needed to estimate $g_{K} (0)$.

The general expectation is that there are other Lehmer pairs 
that produce bounds for $\Lambda$ that are even closer to $0$
(see the discussion in Csordas, Smith, and Varga [4] and
Odlyzko [8]).  However, at this time we do not know of
another pair that is likely to produce a better bound.  The
computations of van de Lune, te Riele, and Winter [5] 
do not prove conclusively that there is no closer pair
among the first $1.5 \cdot 10^9$ zeros of the zeta function.
However, given the search method used, it seems unlikely that
such a pair was missed.  The computations of Odlyzko [8] 
near the zero $\frac{1}{2} + i \gamma_{m}$ of the 
$\zeta$-function, with $m = 10^{20}$, as well as in some other high
intervals, did find some close Lehmer pairs, but none of them seem
to lead to results as good as we obtain here.

The proof of the next lemma is patterned after Lemma 5.1 of Csordas, 
Smith, and Varga [4].  

\begin{Lemma}
Suppose $\Lambda < 0$. Then, the pair of consecutive simple positive 
zeros $\left\{ x_{K} (0); x_{K+1}(0)\right\}$ in (2.5) 
is a Lehmer pair of zeros of $H_{0}$. 
\end{Lemma}

\noindent
{\em Proof}.
We first establish an upper bound for $g_{K} (0)$ of (1.12) for 
$K := 1,048,449,114$. Writing for convenience $x_{j} := x_{j} (0), ~ g_{K} (0)$ 
can be expressed as the sum of the following three terms: 
\renewcommand{\theequation}{2.6}
\be
g_{K} (0) = M_{K,n} + I_{K,n+1}+R_{K,K+n+2}, {\rm \ where\ } n := 9,998,
\ee
and where
\[
\begin{array}{rcl}
M_{K,n} & := & \dis{\sum_{\stackrel{j=K-n}{j\neq K, K+1}}^{K+n+1}} ~ 
\left\{ \frac{1}{(x_{K}-x_{j})^{2}} + \frac{1}{(x_{K+1}-x_{j})^{2}}\right\}, \\
 & & \\
 I_{K,n+1} & := & \dis{\sum_{j=-K-n-1}^{K-n-1}}' ~ 
 \left\{ \frac{1}{(x_{K}-x_{j})^{2}} + 
 \frac{1}{(x_{K+1}-x_{j})^{2}}\right\},
\end{array}
\]
and 
\[
R_{K,K+n+2} := \dis{\sum_{|j| \geq K+n+2}} ~ 
\left\{ \frac{1}{(x_{K}-x_{j})^{2}} + 
\frac{1}{(x_{K+1}-x_{j})^{2}}\right\}.
\]
We separately bound the sums $M_{K,n}, I_{K,n+1}$, and 
$R_{K,K+n+2}$.  

Consider first $M_{K,n}$. Since $\Lambda < 0$ by hypothesis, it follows that 
all the zeros of $H_{0}$ are real and simple (cf.\ Lemma 2.2 of Csordas, Smith, 
and Varga [4]). Hence, from the definition of $M_{K,n}$, 
\begin{eqnarray*}
M_{K,n} &<& \dis{\sum_{\stackrel{j = K-n}{j \neq K, K+1}}^{K+n+1}}
\left\{ \frac{1}{(x_{K}-x_{K-1})^{2}} + 
\frac{1}{(x_{K+1}-x_{K+2})^{2}}\right\} \\
 &=& 2n \left\{ \frac{1}{(x_{K}-x_{K-1})^{2}} + 
\frac{1}{(x_{K+1} - x_{K+2})^{2}}\right\}, 
\end{eqnarray*}
so that
\renewcommand{\theequation}{2.7}
\be
 M_{K,n} < \dis{\frac{n}{2} \left\{ 
 \frac{1}{(\gamma_{K} - \gamma_{K-1})^{2}} + 
 \frac{1}{(\gamma_{K+2}-\gamma_{K+1})^{2}}\right\}}.
\ee
Now, the newly computed zeros, $\gamma_{K-1}$ and $\gamma_{K+2}$, along 
with $\gamma_{K}$ and $\gamma_{K+1}$, are 
\renewcommand{\theequation}{2.8}
\be
\left\{ \begin{array}{rcl}
\gamma_{K-1} & =  & 3.888~588~853~843~374~083\cdot 10^{8},  \\
\gamma_{K} & =  & 3.888~588~860~022~851~203\cdot 10^{8}, \\
\gamma_{K+1} & =  & 3.888~588~860~023~936~899\cdot 10^{8},  \\
\gamma_{K+2} & =  & 3.888~588~866~907~450~543\cdot 10^{8}. 
\end{array} \right.
\ee
Thus, with the above numbers and with $n := 9,998$, 
the upper bound of (2.7), when rounded upward to the next integer, becomes  
\renewcommand{\theequation}{2.9}
\be
M_{K,n} < 23,642.
\ee

We next bound above $I_{K,n+1}$ by 
\renewcommand{\theequation}{2.10}
\begin{eqnarray}
I_{K,n+1} & < 2 & \dis{\sum_{j=-K-n-1}^{K-n-1}}\!\!\!\!' ~ 
\frac{1}{(x_{K}-x_{K-n-1})^{2}} \\
          & = & \frac{4K}{(x_{K}-x_{K-n-1})^{2}} \nonumber \\
          & = & \frac{K}{(\gamma_{K}-\gamma_{K-n-1})^{2}}. \nonumber 
\end{eqnarray}
With the value of $\gamma_{K}$ from (2.8) and with our
calculated value of 
\[
\gamma_{K-n-1} = \gamma_{K-9999} = 3.888~553~840~902~274~209\cdot 10^{8},
\]
the upper bound of (2.10), when rounded upward to the next integer, is 
\renewcommand{\theequation}{2.11}
\be
I_{K,n+1} < 86.
\ee

Finally, we bound above $R_{K,K+n+2}$. Since $H_{0}$ is an even function, 
we have (cf.\ (1.9)) $x_{-j} (0) = -x_{j}(0)$, so that 
$R_{K,K+n+2}$ can be expressed as 
\renewcommand{\theequation}{2.12}
\begin{eqnarray}
&& ~~~R_{K,K+n+2} = \\
&&\dis{\sum_{j = K+n+2}^{\infty}} ~ 
\left\{ \frac{1}{(x_{K}-x_{j})^{2}} + 
\frac{1}{(x_{K+1}-x_{j})^{2}} + 
\frac{1}{(x_{K}+x_{j})^{2}} + 
\frac{1}{(x_{K+1}+x_{j})^{2}}\right\}. \nonumber
\end{eqnarray}
Since $\frac{1}{(x_{K}-x_{j})^{2}} = 
\frac{x_{j}^{2}}{(x_{K}-x_{j})^{2}} \cdot \frac{1}{x_{j}^{2}}$, 
where $\frac{x_{j}^{2}}{(x_{K}-x_{j})^{2}}$ is monotone 
decreasing for $j \geq K + n+2$, the sum of the first term from the 
bracketed quantity in 
(2.12) is bounded above by 
$\frac{x_{K+n+2}^{2}}{(x_{K}-x_{K+n+2})^{2}} \cdot
\dis{\sum_{j=K+n+2}^{\infty}} ~ 
\frac{1}{x_{j}^{2}}$, and the sum of the third term from the 
bracketed quantity in 
(2.12) is bounded above simply by 
$\dis{\sum_{j=K+n+2}^{\infty}} ~ \frac{1}{x_{j}^{2}}$. 
With an analogous treatment for the remaining terms from the bracketed 
quantity in (2.12), we thus have 
\renewcommand{\theequation}{2.13}
\be
~~~~R_{K,K+n+2} < 
\left\{ \frac{x_{K+n+2}^{2}}{(x_{K}-x_{K+n+2})^{2}} + 
\frac{x_{K+n+2}^{2}}{(x_{K+1}-x_{K+n+2})^{2}} + 2
\right\} \cdot \dis{\sum_{j=K+n+2}^{\infty}} 
\frac{1}{x_{j}^{2}}. 
\ee
With the values of $\gamma_{K}$ and $\gamma_{K+1}$ from 
(2.8), and with the calculated value of 
\renewcommand{\theequation}{2.14}
\be
\gamma_{K+n+2} = \gamma_{K+10,000} = 
3.888~623~880~181~523~962\cdot 10^{8},
\ee
we find that
\[
\left\{ \frac{x_{K+n+2}^{2}}{(x_{K}-x_{K+n+2})^{2}} + 
\frac{x_{K+n+2}^{2}}{(x_{K+1}-x_{K+n+2})^{2}} + 
 2 \right\} = 2.465~957\ldots \cdot 10^{10}.
\]
Also, since $\gamma_{K+10,000}$ from (2.14) satisfies 
$[\gamma_{K+10,000}] > 3\cdot 10^{8}$, applying Lemma 2.2 gives
\[
\dis{\sum_{j=K+n+2}^{\infty}} ~ \frac{1}{x_{j}^{2}} \leq 
\frac{\log ([\gamma_{K+n+2}]-1)+1}{4([\gamma_{K+n+2}]-1)} = 
1.335~866~927\ldots \cdot 10^{-8}.
\]
Substituting in the right side of (2.13) then gives, on rounding 
upward to the next integer, 
\renewcommand{\theequation}{2.15}
\be
R_{K, K+n+2} < 330.
\ee

Combining the upper estimates of (2.9), (2.11), and 
(2.15) gives
\renewcommand{\theequation}{2.16}
\be
g_{K} (0) < 24,058.
\ee
But $\Delta_{K} := x_{K+1}-x_{K} = 
2\left(\gamma_{K+1} - \gamma_{K}\right)$, 
so (2.8) gives
\renewcommand{\theequation}{2.17}
\be
\Delta_{K} = 2.171~392\ldots~ \cdot 10^{-4},
\ee
and with (2.16), we then have
\[
\Delta_{K}^{2} \cdot g_{K} (0) < 1.134~321\ldots \cdot 10^{-3} < 4/5.
\]
Thus from (1.11) of Definition 1.1, 
$\{ x_{K} ; x_{K+1}\}$ is a Lehmer pair of zeros of $H_{0}$.\proofend

Finally, we establish our new result, Theorem 2.2 below. If 
$\Lambda \geq 0$, the lower bound of (2.18) is trivially true. Hence, 
assume, as in Lemmas 2.2 and 2.3, that $\Lambda < 0$. We note that $\lambda_{k}$, 
as defined in (1.13), is a 
monotone decreasing function of $g_{k} (0)$ (if $\Delta_{k}^{2} \cdot g_{k} 
(0) < 4/5$). Hence the upper bound for $g_{K} (0)$ in (2.16),
when used to determine $\lambda_{k}$ in (1.13), gives the 
lower bound $-5.895 \cdot 10^{-9}$ of (2.18) for $\Lambda$, as claimed 
in the Abstract above. 


\begin{Theorem}
A lower bound for the de Bruijn-Newman constant $\Lambda$ 
 is
\renewcommand{\theequation}{2.18}
\be
-5.895\cdot 10^{-9} < \Lambda.
\ee
\end{Theorem}

As remarked in Csordas, Smith, and Varga [4], the lower bound for 
$\Lambda$ in (2.18) is quite {\em insensitive\/} to upper estimates of 
$g_{K} (0)$. This can be seen from the following Taylor series of 
$\lambda_{K}$ of (1.13), in terms of $\Delta_{K}^{2} g_{K} (0)$ and its powers:
\renewcommand{\theequation}{2.19}
\be
\lambda_{K} = - ~ \frac{\Delta_{K}^{2}}{8} - 
\frac{\Delta_{K}^{4} g_{K} (0)}{64} - 
\frac{\Delta_{K}^{6} g_{K}^{2} (0)}{128} - 
\frac{11 \Delta_{K}^{8} g_{K}^{3} (0)}{2048} - \cdots ~ ,
\ee
where we note, from (2.17), that just the first term of (2.19) is
\[
- ~\frac{\Delta_{K}^{2}}{8} = - ~5.893~679\ldots \cdot 10^{-9}.
\]

\section{Appendix: Proof of Lemma 2.1}
By (2.1), we can write $N(T) = s(T) + e(T)$, where 
\renewcommand{\theequation}{3.1}
\be
s(T) := \left( \frac{T}{2\pi}\right) \log 
\left(\frac{T}{2\pi}\right) - \frac{T}{2\pi} + \frac{7}{8} ~~~~~ 
(T\geq 2).
\ee
Then
\[
s(T+1)-s(T)=\left(\frac{T+1}{2\pi}\right) \log 
\left(\frac{T+1}{2\pi}\right) - \left(\frac{T+1}{2\pi}\right) - 
\left(\frac{T}{2\pi}\right) \log \left(\frac{T}{2\pi}\right) + 
\left(\frac{T}{2\pi}\right).
\]
Writing $\log \left(\frac{T+1}{2\pi}\right) = \log 
\left(\frac{T}{2\pi}\right) + \log 
\left( 1 + \frac{1}{T}\right) = \log 
\left(\frac{T}{2\pi}\right) + \frac{1}{T} - \frac{1}{2T^{2}} + 
\frac{1}{3T^{3}} - \cdots~$,
we find that
\[
\begin{array}{rcl}
s(T+1) - s(T) & = & \frac{\log T-\log 2\pi}{2\pi} + 
\frac{1}{2\pi} \left\{ 1+\frac{1}{2T} - 
\frac{1}{6T^{2}} + \frac{1}{12T^{3}} - \cdots\right\} - 
\frac{1}{2\pi} \\
& & \\
 & = & \frac{\log T - \log 2\pi}{2\pi} + \frac{1}{2\pi}
\left\{ \frac{1}{2T} - \frac{1}{6T^{2}} + 
\frac{1}{12T^{3}} - \cdots \right\} < 
\frac{\log T-\log 2\pi}{2\pi} + \frac{1}{4\pi T},
\end{array}
\]
where the upper bound arises from taking the first term of the alternating 
series above.
Hence by (2.2),
\[
\begin{array}{rcl}
N(T+1) - N(T) & < & \dis{\frac{\log T}{2\pi} - 
\frac{\log 2\pi}{2\pi} + \frac{1}{4\pi T} + 
|e(T+1)| + | e(T)|} \\
& & \\
 & < & \dis{\frac{\log T}{2\pi} - 
 \frac{\log 2\pi}{2\pi} + \frac{1}{4\pi T} + 
 0.137 \left( 2 \log T + \log \left(1+\frac{1}{T}\right)\right)} \\
 & & \\
 & + & \dis{0.886 \log \log (T+1) + 8.70 ~~~~~ (T \geq 2)}.
\end{array}
\]
Using the upper bound $\log (1+\frac{1}{T}) < \frac{1}{T}$ and evaluating the
constants, this gives 
\renewcommand{\theequation}{3.2}
\be
\begin{array}{rcl}
N(T+1) - N(T) & < & 0.433~154~943 \log T + 0.886 \log \log 
(T+1) + 8.407~492~780  \\
& & \\
 & & + ~ \dis{\frac{0.216~577~472}{T}}.
\end{array}
\ee
It can be easily seen that
\renewcommand{\theequation}{3.3}
\be
0.886 \log \log (T+1) \leq \alpha \log T 
{\rm \ for\ } \alpha := 0.134~874~935 ~~ (T \geq 3 \cdot 10^{8}),
\ee
and
\renewcommand{\theequation}{3.4}
\be
8.407~492~780 \leq \beta \log T {\rm \ for\ } 
\beta := 0.430~727~320 ~~~~~ (T\geq 3\cdot 10^{8}).
\ee
Thus, inserting the bounds of (3.3) and (3.4) 
in (3.2) gives
\renewcommand{\theequation}{3.5}
\be
 ~~~~~ N(T+1)-N(T) < 0.998~757~198 \log T + 
\frac{0.216~577~472}{T} < \log T ~ (T \geq 3\cdot 10^{8}),
\ee
which is the desired result of (2.3) of Lemma 2.1. \proofend

\bigskip
\noindent
{\bf Acknowledgement.} The authors are grateful to H. J. J. te Riele
for pointing out the errors contained in the original version of this
manuscript.
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\end{document}