A note on the real part of the Riemann zeta-function
Juan Arias de Reyna
We consider the real part $\Re\zeta(s)$ of the Riemann zeta-function $\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $\sigma_0 \approx 1.19$ which is defined to be the supremum of $\sigma$ such that $\Re\zeta(\sigma+it)$ can be negative (or zero) for some real $t$. We also consider intervals where $\Re\zeta(1+it) \le 0$ and show that they are rare. The first occurs for $t \approx 682112.9$, and has length $\approx 0.05$. We list the first $50$ such intervals.
1112.4910.tex
% Contribution to Herman te Riele's retirement "liber amicorum"
% By Juan Arias de Reyna, Richard P. Brent and Jan van de Lune
%
% Draft #10 (final), last revised 20111012
%
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\begin{document}

   \author{Juan Arias de Reyna}
   \address{Universidad de Sevilla, Facultad de Matem\'aticas,
	    Apdo.\ 41080-Sevilla, Spain}
   \email{arias@us.es}
   
   \author{Richard P. Brent}

   \address{Mathematical Sciences Institute, 
   Australian National University, Canberra, ACT 0200, Australia}
   \email{abl@rpbrent.com}

   \author{Jan van de Lune }
   \address{\noindent Langebuorren 49, 9074 CH Hallum, The Netherlands 
   \newline(Formerly at CWI, Amsterdam ) }
   \email{j.vandelune@hccnet.nl}

    % title

   \title[On the real part of the Riemann zeta-function]
   {A note on the real part of the Riemann zeta-function}

   % Note that the short title for running heads goes in square
   % brackets.  This is optional.  The long title goes in curly
   % braces.  In the long title, line breaks are indicated by \\.

   \begin{abstract}
   We consider the real part $\Re\zeta(s)$ of the Riemann zeta-function
   $\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute
   accurately the constant $\sigma_0 \approx 1.19$ 
   which is defined to be the supremum of $\sigma$ such
   that $\Re\zeta(\sigma+it)$ can be negative (or zero)
   for some real $t$.
   We also consider intervals where $\Re\zeta(1+it) \le 0$ 
   and show that they are rare. The first occurs for $t \approx 682112.9$,
   and has length $\approx 0.05$.  
   We list the first $50$ such intervals.
   \end{abstract}

\vspace*{-10pt}

   % AMS subject classifications (used in AMS journals)
   %\subjclass{Primary 00A30; Secondary 00A22, 03E20}

   % AMS keywords (used in AMS journals)
   %\keywords{Bernoulli numbers, integers.}

   % acknowledge support, etc
   %\thanks{}

   % dedication
   \dedicatory{Dedicated to Herman J.~J.~te Riele on the occasion
     of his retirement\\ from the CWI in January 2012}

   % today's date, or fill in whatever date you prefer
   %\date{\today}


   \maketitle

\vspace*{-10pt}

\section{Introduction}

   In this note we consider the real part of the Riemann zeta-function
   $\zeta(s)$ in the half-plane $H = \{s \in \C\,|\,\Re(s) \ge 1\}$. 
   As usual, we write
   $s = \sigma + it$, so $\Re(s) = \sigma \ge 1$. We are mainly interested
   in the regions where $\Re\zeta(s) \le 0$.
   Since $\lim_{\sigma\uparrow\infty}\zeta(\sigma+it)=1$ 
   (uniformly in $t$), $\Re\zeta(\sigma+it)$
   cannot be zero for arbitrarily large $\sigma>1$.
   We define \[\sigma_0 := \sup \{\sigma \in \R\,|\, (\exists t \in \R)\,
		\Re\zeta(\sigma+it) = 0\}.\]

   Thus, $\Re\zeta(s) > 0$ if $\sigma > \sigma_0$.
   In van de Lune \cite{L} it was shown that $\sigma_0$
   is the (unique) positive real root of the equation
   \begin{displaymath}
   \sum_p\arcsin\Bigl(\frac{1}{p^\sigma}\Bigr) = \frac{\pi}{2}\,,
   \end{displaymath}
   where $p$ runs through the primes (we adopt this convention throughout).
   In \cite{L} it was also shown that $\sigma_0>1.192$ and that
   $\Re\zeta(\sigma_0 + it)$ never vanishes. 

   The main aim of this note is to show how $\sigma_0$ can be computed to 
   arbitrarily high precision by an efficient algorithm. 
   We also mention some results on the behaviour of
   $\Re\zeta(\sigma+it)$ for $1 \le \sigma < \sigma_0$, and in particular
   on the line $\sigma = 1$. 
%  Details of these results will be given
%  in a longer paper which is currently in preparation.

\section{Accurate computation of the constant $\sigma_0$} \label{sec:theory}

   In this section we assume that $\sigma \ge \sigma_1 > 1$, where 
   $\sigma_1$ is a suitable constant (e.g. $1.1$).
   We show how the constant $\sigma_0$ can be computed within a given
   error bound. There are three main steps.

   \begin{enumerate}
   \item Give an algorithm to evaluate the 
   {\em prime zeta-function} \cite{Froberg}
	\[P(\sigma) = \sum_p p^{-\sigma},\] for real $\sigma > 1$.
   \item Using step 1, 
   give an algorithm to evaluate the function $f(\sigma)$ defined by
   \begin{displaymath}
   f(\sigma) = \sum_p\arcsin\Bigl(\frac{1}{p^\sigma}\Bigr) - \frac{\pi}{2}.
   \end{displaymath}
   \item Use a suitable zero-finding algorithm to locate a zero of $f(\sigma)$
   in a (sufficiently small) interval where $f(\sigma)$ changes sign,
   for example $[1.1, 1.2]$.
   \end{enumerate}

   Step 1 is easy.  From the Euler product for $\zeta(\sigma)$ and M\"obius 
   inversion, we have a formula essentially known to Euler \cite[1748]{Euler}:
   % Not clear exactly what Euler knew, but "essentially" should cover it.
   % See email from Juan 20111007
   \begin{equation}
   \label{eq:P}
   P(\sigma) = \sum_{r=1}^\infty\frac{\mu(r)}{r}\log\zeta(r\sigma),
   \end{equation}
   which is valid for $\sigma > 1$
   (see Titchmarsh \cite[eqn.\ (1.6.1)]{T}).
%   In fact, (\ref{eq:P}) gives the analytic continuation of $P(s)$
%   for $\Re(s) > 0$, but
%   we only need (\ref{eq:P}) for real $s \ge \sigma_1$. 
   The series converges rapidly in view of the following Lemma.
   \begin{lemma}	\label{lemma:P}
   For $\sigma \ge 2$, $0 < \log \zeta(\sigma) < 3/2^{\sigma}$
   and $0 < P(\sigma) < 3/2^{\sigma}$.
   \end{lemma}
   \begin{proof}
   For $\sigma \ge 2$, we have
   \[ 0 < \zeta(\sigma) - 1 < 2^{-\sigma} + 3^{-\sigma}
	+ \int_3^\infty x^{-\sigma}{\rm d}x
	= 2^{-\sigma} + 3^{-\sigma} + \frac{3^{1-\sigma}}{\sigma-1}
	< 3/2^{\sigma}, \]
   so
   \[ 0 < \log\zeta(\sigma) < \zeta(\sigma) - 1 < 3/2^{\sigma}.\]
   The upper bound on $P(\sigma)$ follows similarly, using
   $P(\sigma) < \zeta(\sigma) - 1$.
   \end{proof}
   Using (\ref{eq:P}) and Lemma~\ref{lemma:P}, we have
   \[P(\sigma) = \log\zeta(\sigma) +
	\sum_{r=2}^\infty\frac{\mu(r)}{r}\log\zeta(r\sigma),\]
   where the $r$-th term in the sum is bounded in absolute value
   by $3/2^{r\sigma+1}$. Thus, we can evaluate $P(\sigma)$ accurately,
   for given $\sigma > 1$, using any good algorithm for the evaluation of
   $\zeta(\sigma)$, for example Euler-Maclaurin summation.
   If (\ref{eq:P}) is used to compute $P(\sigma)$, $P(3\sigma)$,
   $P(5\sigma), \ldots,$ then we should take care to compute the relevant
   terms $\log\zeta(r\sigma)$ only once.

%   We intend to compute the solution $\sigma_0$ (or $\sigma(1)$) of the
%   equation $\f(\sigma) = 0$ 
%   to arbitrarily high precision. This requires that we are able to compute 
%   $f(\sigma):=\sum_p\arcsin\bigl(\frac{1}{p^\sigma}\bigr)$ for $\sigma>1$
%   with an error $<\varepsilon$. 
   
   For step 2, we observe that the $\arcsin$ series defining $f(\sigma)$
   converges slowly and irregularly, since it is a sum over primes which to
   first order behaves like $\sum_p p^{-\sigma}$.  The well-known ``trick''
   is to express $f(\sigma)$ as a double series and reverse the order of
   summation, obtaining an expression which is mathematically equivalent but
   computationally far superior.  For some similar examples, see 
   Wrench \cite[1961]{Wrench}.
 
   For $|x|<1$ we have 
   \begin{displaymath}
   \arcsin(x)=\sum_{k=0}^\infty c_k x^{2k+1},
   \end{displaymath}
   where 
   \begin{displaymath}
   c_k=\frac{1\cdot3\cdot5\cdots(2k-1)}{2\cdot 4\cdot6\cdots(2k)}
   \frac{1}{2k+1}=\frac{(2k)!}{(2^kk!)^2(2k+1)}\qquad\text{for $k\ge0$}.
   \end{displaymath}
   Note that all $c_k$ are positive so that $f(\sigma)$ is strictly convex.
   It is also clear that $f(\sigma)$ is strictly
   decreasing for $\sigma>1$. 
%   These aspects are of importance for the numerical solution of 
%   $f(\sigma)=A:=\frac{\pi}{2}$. 
%   
%   Indeed, this makes it possible to (safely!) avoid the explicit computation
%   of the derivative $f'(\sigma)$.
   From the expression for $c_k$, we see that, for $k \ge 1$,
   \begin{equation}
   c_k \le \frac{1}{2(2k+1)}\,.			\label{eq:cineq}
   \end{equation} 

   For $\sigma > 1$ it is easy to justify interchanging the
   order of summation in
   \begin{equation*}
   f(\sigma)=\sum_p\sum_{k=0}^\infty c_k\Bigl(\frac{1}{p^\sigma}\Bigr)^{2k+1}
	- \; \frac{\pi}{2}\,,
   \end{equation*}
obtaining
   \begin{equation}		\label{eq:fseries}
   f(\sigma) =\sum_{k=0}^\infty c_k\sum_p\frac{1}{p^{(2k+1)\sigma}}
	- \frac{\pi}{2}
   =\sum_{k=0}^\infty c_k P\bigl((2k+1)\sigma\bigr)
	- \frac{\pi}{2}\,.
 %  &=&\sum_{k=0}^K c_k P\bigl((2k+1)\sigma\bigr)+
 %  \sum_{k=K+1}^\infty c_k P\bigl((2k+1)\sigma\bigr)
 % 	- \frac{\pi}{2}
   \end{equation}

   From Lemma~\ref{lemma:P} and the inequality (\ref{eq:cineq}),
   we see that
   \[
   0 < 
   \sum_{k=K+1}^\infty c_k P\bigl((2k+1)\sigma\bigr) < 2^{-(2K+3)\sigma},
   \]
   so it is easy to determine $K$ such that we can truncate the series
   in (\ref{eq:fseries}) to a finite sum over $k \le K$ with a rigorous
   error bound.

   If desired, we can substitute (\ref{eq:P}) into (\ref{eq:fseries})
   and interchange the order of summation, obtaining

   \begin{equation}	\label{eq:fseries2}
   f(\sigma) = \sum_{j \ge 0} d_j \log \zeta((2j+1)\sigma)
     - \frac{\pi}{2},
   \end{equation}
   where
   \[
   d_j = \sum_{k \ge 0, r \ge 1, (2k+1)r = 2j+1}
		\frac{c_k\mu(r)}{r}\,.
   \]
   From the inequality $c_k \le 1/(2k+1)$ (valid for $k \ge 0$),
   it follows that $|d_j| \le 1$.
   Using Lemma \ref{lemma:P}, we can determine where to safely
   truncate the series (\ref{eq:fseries2}).

   For step 3, we can use a zero-finding algorithm which needs only
   function (not derivative) evaluations, and gives a guaranteed bound
   on the final result.  For example, the method of bisection could be
   used, but would be slow, taking about $\log_2(1/\varepsilon)$
   function evaluations to obtain a solution with error bounded
   by $\varepsilon$.  In the secant method, a sequence $(x_n)$, converging
   to a zero of $f$ under suitable conditions, is obtained by
   computing the approximation
   $x_{n}$ by linear interpolation using the two
   points
   $(x_{n-1}, f(x_{n-1}))$ and $(x_{n-2}, f(x_{n-2}))$. 
   It converges with order
   $(1+\sqrt{5})/2 \approx 1.618$, but does not always give a
   guaranteed bound on the error.  A combination of bisection
   and linear interpolation, as in the algorithms
   of Dekker~\cite{Dekker} or Brent~\cite{Br},
   can give convergence about as fast as the secant method, but 
   with the final result bracketed in a short interval where the function $f$
   changes sign.

   \section{Computational results}

   The second and third authors independently wrote programs implementing
   the ideas of \S\ref{sec:theory}, using Magma in one case and
   Mathematica 4 and 8 in the other case. The programs used different
   strategies to obtain a final interval where $f$ changes sign (in one
   case taking advantage of the strict convexity of $f$). 
   The output of the programs agrees to at least 500D.
% Jan - I don't think my program ever produced a 1000D result,
% due to a problem with Magma.  I will try again.  Richard
   We give here the correctly rounded result to 100D:

   \begin{displaymath}\begin{split}
   \sigma_0\approx
   1.&19234\,73371\,86193\,20289\,75044\,27425\,59788\,34011\,19230\,
   83799\\[-4pt]
   &94301\,37194\,92990\,52458\,64848\,30139\,24084\,99863\,83788\,36244\,.\\
%   1.&19234\,73371\,86193\,20289\,75044\,27425\,59788\,34011\,19230\,
%   83799\\
%   &94301\,37194\,92990\,52458\,64848\,30139\,24084\,99863\,83788\,36243\\
%   &82491\,  94002\,23611\,44827\,89487\,29520\,27840\,08384\,30094\,
%   19813\\
%   &31650\,56133\,51881\,52903\,45892\,10670\,22916\,
%   \cdots
   \end{split}\end{displaymath}
   Programs and higher precision values are available from the authors.
   % See magmarpb/reyna5

   \section{The distribution of $\Re\zeta(\sigma+it)$ for $\sigma \ge 1$}

   Assuming that the limit exists, we define
   \[
   d(\sigma) = \lim_{T \to +\infty}
		\frac{1}{T}\,m\{t\in [0,T]\,|\, \Re\zeta(\sigma + it) < 0\},
   \]
   where $m$ denotes Lebesgue measure.
%
%\noindent{\bf Note (RPB)}: 
%    I think that the existence of the limit follows from the
%    result of Bohr and Jessen \cite{BJ} quoted in Titchmarsh
%    \cite[\S11.13]{T}. This seems straightforward for $\sigma > 1$ 
%    because then $\arg\zeta(\sigma + it)$ is bounded and 
%    \begin{eqnarray*}
%    &&\{ t \in [-T,T] \,:\, \Re\zeta(\sigma + it) < 0\} =\\
%    &&\{t \in [-T, T] \,:\, |\Im\log\zeta(\sigma + it)| \in
%       (\pi/2, 3\pi/2) \cup (5\pi/2, 7\pi/2) \cup \cdots \} 
%    \end{eqnarray*} 
%    but $|\Im\log\zeta(\sigma + it)|$ is bounded so we only need to take
%    a finite union of intervals for $\Im\log\zeta(\sigma+it)$,
%    and there is no restriction on $\Re\log\zeta(\sigma+it)$,
%    so we have a finite union of rectangles $R$ as in Titchmarsh,
%    and the limit exists by using the Theorem of Bohr and Jessen on each
%    rectangle. In the case $\sigma = 1$ we can take say
%    the first $k$ intervals to get a lower bound $\delta_k$; the 
%    $\delta_k$ form a monontonic increasing sequence that is bounded,
%    so has a limit.\\
%   {\bf End Note}
%
   Informally, $d(\sigma)$ is the probability that $\zeta(s)$ has
   negative real part on a given vertical line $\Re(s) = \sigma$.

   The results of Section \ref{sec:theory} show that
   $d(\sigma) = 0$ for $\sigma \ge \sigma_0 \approx 1.19$.
   Here we briefly discuss the region $1 \le \sigma < \sigma_0$.

   At least for those values of $t$ that are accessible to computation, 
   $\Re\zeta(\sigma+it)$ is ``usually'' positive for $\sigma \ge 1$.
   The function
   $d(\sigma)$ is conjectured to be continuous and monotonic decreasing
   from a positive value at $\sigma = 1$ to zero at $\sigma = \sigma_0$.
   Even on the line $\sigma = 1$, $\Re\zeta(\sigma+it)$ is usually positive
   \cite{Milioto}.
   We can prove that $d(1) < 1/4$, but a Monte Carlo computation
   suggests that the true value is much smaller. Based on $5 \times 10^{11}$
   pseudo-random trials, we estimate $d(1) = (3.80 \pm 0.01) \times 10^{-7}.$
   Similarly, we estimate
   $d(1.01) = (1.10\pm 0.01) \times 10^{-7}$ and
   $d(1.02) \approx (2.66\pm 0.04) \times 10^{-8}$, so it can be seen that 
   $d(\sigma)$ decreases rapidly as we move to the right of $\sigma = 1$.

   Although $\zeta(s)$ has a simple pole at $s = 1$, the Laurent series
  \[\zeta(s) = \frac{1}{s-1} + \gamma + O(|s-1|)\]
   shows that $\Re\zeta(1+it)$ % is bounded for small (real) $t$, and 
   has a positive limit $\gamma = 0.577\cdots$ (Euler's constant) 
   as $t \to 0$.

   On any fixed vertical line $\sigma > 1$, both
   $\zeta(\sigma+it)$ and $1/\zeta(\sigma+it)$ are bounded,
   in fact
   ${\zeta(2\sigma)}/{\zeta(\sigma)} < |\zeta(\sigma + it)|
	\le \zeta(\sigma)$.
   However, the situation is different on the line $\sigma = 1$,
   as both $\zeta(1+it)$ and $1/\zeta(1+it)$ are unbounded. Their 
   true order of growth is unknown.
%   but Titchmarsh~\cite[\SS3.5--3.6]{T} gives the estimates
%   \[\zeta(1+it) = O(\log t), \;\; 1/\zeta(1+it) = O(\log^7 t)\]
%   for $t \ge 2$, and also~\cite[Thm 8.5]{T}
%   \[\zeta(1+it) = \Omega(\log\log t).\]
   It follows from Titchmarsh~\cite[Theorem 11.9]{T} and the continuity
   of $\Re\zeta(1+it)$ that
% the range of the function $\Re\zeta(1+it)$ is $\R$, that is to say that 
   $\Re\zeta(1+it)$ attains all real values. 
   Nevertheless, the ``usual''
   values are quite small. As a special case of
   \cite[Theorem 7.2]{T} we have the mean value theorem
   \[\lim_{T \to \infty} \frac{1}{T}\,\int_1^T |\zeta(1+it)|^2\, {\rm d}t
     = \zeta(2) = \frac{\pi^2}{6}\,.\] 
   Using ideas as in the proof of \cite[Theorem 7.2]{T},
   % See Juan's 110927-Question.pdf 
   we can prove that
   \[\lim_{T \to \infty} \frac{1}{T}\,\int_0^T \Re\zeta(1+it)\, {\rm d}t
     = 1\,.\] 
   Thus, informally, we can say that the typical value of $\Re\zeta(1+it)$
   is close to~$1$. The values have a distribution with mean $1$
   and variance $\pi^2/6 - 1 \approx 0.645$.

   In~\cite[Table 1]{L}, van de Lune gave a list of values 
   of $t > 0$ such that
   $\Re\zeta(1+it) < 0$ and is (approximately) a local minimum. 
   The list was not claimed to be exhaustive.  The smallest $t$ listed
   was $t = 682112.92$ with $\Re\zeta(1+it) \approx -0.003$. We have shown,
   using the ``maximum slope principle'' \cite{LR}, 
   that this is very close to the smallest $t$ for which 
   $\Re\zeta(1+it) \le 0$.  More precisely,
   $\Re\zeta(1+it) > 0$ for $0 < t < 682112.8913$,
   and there is a local minimum of $-.0027652$ at $t \approx 682112.9169$.
   In applying the maximum slope principle we used the bound
   % See Juan's Inequality.pdf
   \[
    \left|\frac{\rm d}{{\rm d}t}\arg\zeta(1+it)\right| =
    \left|\Re\frac{\zeta'(1+it)}{\zeta(1+it)}\right| \le
    \frac{3}{4}\,\log(t^2+4) + 7 \;\; {\rm for} \;\; t \ge 10. \]

   \begin{table}[tp] % generated using awk from intervals-01-50.txt
   \centering
   \caption{First $50$ negative local minima of $\Re\zeta(1+it)$}
   \label{table:one}
   \begin{tabular}{|c|c|c||c|c|c|}
   \hline
   $t$ & $\Re\zeta$ & length &
   $t$ & $\Re\zeta$ & length \\
   \hline
    $\;\;$
   $  682112.9169$ & $-0.0028$ & $0.0529$&
   $ 8350473.4853$ & $-0.0019$ & $0.0451$\\
   $ 1267065.1710$ & $-0.0040$ & $0.0655$&
   $ 8366684.0439$ & $-0.0197$ & $0.1322$\\
   $ 1466782.0667$ & $-0.0013$ & $0.0391$&
   $ 8452317.9526$ & $-0.0090$ & $0.0900$\\
   $ 1858650.0915$ & $-0.0282$ & $0.1686$&
   $ 8967566.5926$ & $-0.0148$ & $0.1336$\\
   $ 2023654.7671$ & $-0.0221$ & $0.1389$&
   $ 9960968.8748$ & $-0.0184$ & $0.1373$\\
   $ 2064996.2141$ & $-0.0117$ & $0.1076$&
   $11231380.7309$ & $-0.0099$ & $0.1042$\\
   $ 2195056.7909$ & $-0.0755$ & $0.2718$&
   $11236680.3350$ & $-0.0262$ & $0.1595$\\
   $ 2202620.3296$ & $-0.0111$ & $0.1159$&
   $11781932.0257$ & $-0.0170$ & $0.1288$\\
   $ 2530662.6360$ & $-0.0072$ & $0.0865$&
   $11884021.9776$ & $-0.0035$ & $0.0564$\\
   $ 3259774.5293$ & $-0.0471$ & $0.2098$&
   $12045289.3337$ & $-0.0644$ & $0.2498$\\
   $ 3548283.4160$ & $-0.0189$ & $0.1459$&
   $12276788.1573$ & $-0.0182$ & $0.1476$\\
   $ 4052438.9330$ & $-0.0023$ & $0.0474$&
   $12546625.7916$ & $-0.0455$ & $0.2031$\\
   $ 4197235.0783$ & $-0.0331$ & $0.1977$&
   $12781127.5748$ & $-0.0102$ & $0.0964$\\
   $ 5410820.7150$ & $-0.0008$ & $0.0307$&
   $13598773.5889$ & $-0.0543$ & $0.2317$\\
   $ 6027913.8513$ & $-0.0181$ & $0.1325$&
   $13786262.5457$ & $-0.0826$ & $0.2635$\\
   $ 6164063.0008$ & $-0.0263$ & $0.1603$&
   $13922411.7750$ & $-0.0222$ & $0.1418$\\
   $ 6238849.4877$ & $-0.0071$ & $0.0827$&
   $14190358.4974$ & $-0.0632$ & $0.2214$\\
   $ 6265907.4688$ & $-0.0030$ & $0.0522$&
   $14391623.0217$ & $-0.0016$ & $0.0437$\\
   $ 6421627.2235$ & $-0.0241$ & $0.1651$&
   $14788310.5330$ & $-0.0149$ & $0.1132$\\
   $ 7338152.4379$ & $-0.0043$ & $0.0656$&
   $14856540.3430$ & $-0.0220$ & $0.1442$\\
   $ 7469838.9709$ & $-0.0009$ & $0.0305$&
   $15173904.7533$ & $-0.0041$ & $0.0800$\\
   $ 7766995.0303$ & $-0.0742$ & $0.2840$&
   $15321273.7219$ & $-0.0131$ & $0.1181$\\
   $ 7774558.3985$ & $-0.0672$ & $0.2705$&
   $16083163.0244$ & $-0.0098$ & $0.1038$\\
   $ 7985493.9836$ & $-0.0324$ & $0.1728$&
   $16503899.3235$ & $-0.0060$ & $0.0680$\\
   $ 8299958.2327$ & $-0.0022$ & $0.0432$&
   $16656258.8346$ & $-0.0155$ & $0.1329$\\
   \hline
   \end{tabular}
   \end{table}

   Table~\ref{table:one} lists the first $50$ local
   minima of $\Re\zeta(1+it)$ for which\linebreak 
   $t > 0$ and $\Re\zeta(1+it) \le 0$ (no minima are exactly zero).
   The values  
   in the table are rounded to $4$ decimal places.
   The columns headed ``length'' give the lengths of the intervals 
   containing $t$
   in which $\Re\zeta$ is negative. To $8$ decimal places, the first interval,
   of length $0.05291225$, is $(682112.89133824, 682112.94425049)$. 
% The # of intervals <= 20 000 000 is 56 ( rigorous ) the last one having  
% t0  >= 19 593 992.
%  In the range $0 < t \le 2 \times 10^7$ there are $56$ such intervals. 
   The sum of the lengths of the first $50$ intervals is 
   $6.48390168$, % 6.4839016815353441697 according to Juan
   giving an estimate $d(1) \approx 3.85\times 10^{-7}$.
% To estimate d(1) I use (sum lengths - 0.5*(last length))/(last t)
% Simply (sum lengths)/(last t) is biased by being taken just after a 
% local minimum.
   This is close to our Monte Carlo 
   estimate $d(1) \approx 3.80 \times 10^{-7}$.
%  especially as ``small'' $t$ may be atypical.

   In this brief note we refrain from commenting on the region\linebreak
   $\sigma \in [1/2, 1)$, but refer the interested reader to the literature, 
   such as Bohr and Jessen \cite{BJ}, Titchmarsh \cite[\S11.13]{T},
   Tsang \cite{Tsang}, Joyner \cite{Joyner}, 
   Laurin\v{c}ikas \cite{Laurincikas}, Steuding \cite{Steuding}
   and K\"uhn \cite{Kuhn}.

   \begin{thebibliography}{99}

   \bibitem{BJ}
   \textsc{H.~Bohr and B.~Jessen}, % Harald Bohr, B\"orge Jessen
   \emph{\"Uber die Werteverteilung der Riemannschen Zetafunktion},
   Acta Mathematica {\bf{54}} (1930), 1--35; % I:  sigma > 1
   {\em ibid} {\bf{58}} (1931), 1--55.	     % II: sigma in [1/2, 1]
   % Year of part II given as 1932 in Titchmarsh, but printed 24 Nov 1931

   \bibitem{Br}
   \textsc{R.~P.~Brent},
   \emph{Algorithms for Minimization without Derivatives},
   Prentice-Hall, New Jersey, 1973. Reprinted by Dover, New York, 2002.

%   \bibitem{Dejon}
%   \textsc{B.~Dejon and P.~Henrici (eds.)},
%   \emph{Constructive Aspects of the Fundamental Theorem of Algebra},
%   Interscience, New York, 1969.

   \bibitem{Dekker}
   \textsc{T.~J.~Dekker},
   \emph{Finding a zero by means of successive linear interpolation},
   in \emph{Constructive Aspects of the Fundamental Theorem of Algebra}
   (B.~Dejon and P.~Henrici, eds.), Interscience, New York, 1969.

   \bibitem{Euler}
    \textsc{L.~Euler}, 
    \emph{Introduction \`a l'Analyse Infinit\'esimale}, Vol.~1,
    (original 1748, French translation 1796).
%% Reference from
%% http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html
%% which says that Euler computed P(2), P(4), ... , P(36) to about 13-digit
%% accuracy, presumably using his known values of zeta(2), zeta(4), ... On
%% checking Tome 1, pp 219-220, it seems that Euler did not use the formula
%% involving the Mobius function, just some "ad hoc" methods. This is not
%% surprising, since Mobius (1790-1868) was later than Euler. On the other
%% hand, Juan says: in the book by Narkiewicz The Development of Prime
%% Number Theory in page 239 it is stated that Euler make the assertion
%% sum_n mu(n)/n = 0 .  It is very easy to say somebody has said but not
%% proved. Without noticing that to state this, he must have an idea of what
%% mu(n) was. See Analyse, I, Sec. 268 (page 208 in the 1855 French version).

   \bibitem{Froberg}
   \textsc{C-E.~Fr\"oberg}, %  Fr\"berg, Carl-Erik (1968). 
   \emph{On the prime zeta function}. 
   Nordisk Tidskrift f\"ur Informationsbehandling (BIT) 
   {\bf{8}}, 3 (1968), 187--202.

%   \bibitem{Glaisher}
%   \textsc{J.~W.~L.~Glaisher},  
%    \emph{On the Sums of Inverse Powers of the Prime Numbers}. 
%    Quart. J. Math. {\bf{25}} (1891), 347-–362.
%    Also Merrifield, 1881, see MathWorld or Wikipedia pages

   \bibitem{Joyner}
   \textsc{D.~Joyner},
   \emph{Distribution Theorems of L-functions},
   Pitman Research Notes in Mathematics {\bf{142}},
   John Wiley and Sons, New York, 1986. % MR865983 (88m:11070)
   % For the first published proof of Selberg's theorem on the
   % distribution of log\zeta\(1/2+it), according to Steuding.
   % (But Tsang's thesis was earlier, 1984)

   \bibitem{Kuhn}
   \textsc{P.~K\"uhn}, 	% Patrick K\"uhn
   \emph{On Selberg's central limit theorem},
   Master Thesis, 	% This is how it is spelt on the title page
   Department of Mathematics, ETH Z\"urich, Switzerland,
   March 2011.
%   % http://www.math.ethz.ch/~pdehaye/thesis_students/\
%   % Patrick_Kuhn-Selberg_s_central_limit_theorem.pdf

   \bibitem{Laurincikas}
   \textsc{A.~Laurin\v{c}ikas}, % Antanas Laurin\v{c}ikas
   \emph{Limit theorems for the Riemann zeta-function},
   Mathematics and its Applications {\bf{352}}, 
   Kluwer Academic Publishers, Dordrecht, 1996.    
   % MR1376140 (96m:11070)

   \bibitem{L}
   \textsc{J.~van de Lune}, 
   \emph{Some observations concerning the zero-curves 
   of the real and imaginary parts of Riemann's zeta function}. 
   Afdeling Zuivere Wiskunde [Department of Pure Mathematics], 
   Report ZW 201/83. Mathematisch Centrum, Amsterdam, 1983. i+25 pp.    

   \bibitem{LR}
   \textsc{J.~van de Lune and H.~J.~J.~te~Riele},
   \emph{Numerical computation of special zeros of partial sums of
   Riemann's zeta function},
   Computational Methods in Number Theory, Part II,
   Math.\ Centrum Tracts, Vol.\ 155,
   Math.\ Centrum, Amsterdam, 1982, 371--387.
   % MR702522 (84h:10058)

   \bibitem{Milioto}
   \textsc{D.~C.~Milioto},
   \emph{A method for zeroing-in on $\Re\zeta(\sigma+it) < 0$ in the
   half-plane $\sigma > 1$}, arXiv:1001.2962v3, 20 Jan.\ 2010.
%  This paper is poorly written and  hard to understand. 
%  The first entry in its Table 5 seems to correspond to our
%  t0 = 33926231.46 (interval #89) if we assume k = pi/log(3/2) in
%  the entry 4378640k - 2/5 (which is 33926231.43)

   \bibitem{Steuding}
   \textsc{J.~Steuding},	% J\"orn Steuding
   \emph{Value-distribution of L-functions},
   Lecture Notes in Mathematics {\bf{1877}}, 
   Springer, 2007.
    
   \bibitem{T}
   \textsc{E.~C.~Titchmarsh},
    \emph{The Theory of the Riemann Zeta-function},
   2nd edition, edited by D.\ R.\ Heath-Brown. 
   The Clarendon Press, Oxford, 1986.

   \bibitem{Tsang}
   \textsc{K.~Tsang},	% Kai-man Tsang
   \emph{The Distribution of the Values of the Riemann Zeta-function},
   PhD Thesis, Department of Mathematics, Princeton University, Oct.\ 1984.
   % http://www.math.sjsu.edu/~goldston/Tsang[1-3].pdf for Chapter 1
   % See AMS Notices, 56, 6 (June/July 2009), 692-710 for Tsang's comments
   % on being Selberg's student (the only one?).

   \bibitem{Wrench}
   \textsc{J.~W.~Wrench}, 
   \emph{Evaluation of Artin's constant and the twin prime constant},
   Math.\ Comp.\ {\bf{15}} (1961), 396--398.

   \end{thebibliography}   

\end{document}

1112.4910.extracted_bib
\begin{thebibliography}{99}

   \bibitem{BJ}
   \textsc{H.~Bohr and B.~Jessen}, % Harald Bohr, B\"orge Jessen
   \emph{\"Uber die Werteverteilung der Riemannschen Zetafunktion},
   Acta Mathematica {\bf{54}} (1930), 1--35; % I:  sigma > 1
   {\em ibid} {\bf{58}} (1931), 1--55.	     % II: sigma in [1/2, 1]
   % Year of part II given as 1932 in Titchmarsh, but printed 24 Nov 1931

   \bibitem{Br}
   \textsc{R.~P.~Brent},
   \emph{Algorithms for Minimization without Derivatives},
   Prentice-Hall, New Jersey, 1973. Reprinted by Dover, New York, 2002.

%   \bibitem{Dejon}
%   \textsc{B.~Dejon and P.~Henrici (eds.)},
%   \emph{Constructive Aspects of the Fundamental Theorem of Algebra},
%   Interscience, New York, 1969.

   \bibitem{Dekker}
   \textsc{T.~J.~Dekker},
   \emph{Finding a zero by means of successive linear interpolation},
   in \emph{Constructive Aspects of the Fundamental Theorem of Algebra}
   (B.~Dejon and P.~Henrici, eds.), Interscience, New York, 1969.

   \bibitem{Euler}
    \textsc{L.~Euler}, 
    \emph{Introduction \`a l'Analyse Infinit\'esimale}, Vol.~1,
    (original 1748, French translation 1796).
%% Reference from
%% http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html
%% which says that Euler computed P(2), P(4), ... , P(36) to about 13-digit
%% accuracy, presumably using his known values of zeta(2), zeta(4), ... On
%% checking Tome 1, pp 219-220, it seems that Euler did not use the formula
%% involving the Mobius function, just some "ad hoc" methods. This is not
%% surprising, since Mobius (1790-1868) was later than Euler. On the other
%% hand, Juan says: in the book by Narkiewicz The Development of Prime
%% Number Theory in page 239 it is stated that Euler make the assertion
%% sum_n mu(n)/n = 0 .  It is very easy to say somebody has said but not
%% proved. Without noticing that to state this, he must have an idea of what
%% mu(n) was. See Analyse, I, Sec. 268 (page 208 in the 1855 French version).

   \bibitem{Froberg}
   \textsc{C-E.~Fr\"oberg}, %  Fr\"berg, Carl-Erik (1968). 
   \emph{On the prime zeta function}. 
   Nordisk Tidskrift f\"ur Informationsbehandling (BIT) 
   {\bf{8}}, 3 (1968), 187--202.

%   \bibitem{Glaisher}
%   \textsc{J.~W.~L.~Glaisher},  
%    \emph{On the Sums of Inverse Powers of the Prime Numbers}. 
%    Quart. J. Math. {\bf{25}} (1891), 347-–362.
%    Also Merrifield, 1881, see MathWorld or Wikipedia pages

   \bibitem{Joyner}
   \textsc{D.~Joyner},
   \emph{Distribution Theorems of L-functions},
   Pitman Research Notes in Mathematics {\bf{142}},
   John Wiley and Sons, New York, 1986. % MR865983 (88m:11070)
   % For the first published proof of Selberg's theorem on the
   % distribution of log\zeta\(1/2+it), according to Steuding.
   % (But Tsang's thesis was earlier, 1984)

   \bibitem{Kuhn}
   \textsc{P.~K\"uhn}, 	% Patrick K\"uhn
   \emph{On Selberg's central limit theorem},
   Master Thesis, 	% This is how it is spelt on the title page
   Department of Mathematics, ETH Z\"urich, Switzerland,
   March 2011.
%   % http://www.math.ethz.ch/~pdehaye/thesis_students/\
%   % Patrick_Kuhn-Selberg_s_central_limit_theorem.pdf

   \bibitem{Laurincikas}
   \textsc{A.~Laurin\v{c}ikas}, % Antanas Laurin\v{c}ikas
   \emph{Limit theorems for the Riemann zeta-function},
   Mathematics and its Applications {\bf{352}}, 
   Kluwer Academic Publishers, Dordrecht, 1996.    
   % MR1376140 (96m:11070)

   \bibitem{L}
   \textsc{J.~van de Lune}, 
   \emph{Some observations concerning the zero-curves 
   of the real and imaginary parts of Riemann's zeta function}. 
   Afdeling Zuivere Wiskunde [Department of Pure Mathematics], 
   Report ZW 201/83. Mathematisch Centrum, Amsterdam, 1983. i+25 pp.    

   \bibitem{LR}
   \textsc{J.~van de Lune and H.~J.~J.~te~Riele},
   \emph{Numerical computation of special zeros of partial sums of
   Riemann's zeta function},
   Computational Methods in Number Theory, Part II,
   Math.\ Centrum Tracts, Vol.\ 155,
   Math.\ Centrum, Amsterdam, 1982, 371--387.
   % MR702522 (84h:10058)

   \bibitem{Milioto}
   \textsc{D.~C.~Milioto},
   \emph{A method for zeroing-in on $\Re\zeta(\sigma+it) < 0$ in the
   half-plane $\sigma > 1$}, arXiv:1001.2962v3, 20 Jan.\ 2010.
%  This paper is poorly written and  hard to understand. 
%  The first entry in its Table 5 seems to correspond to our
%  t0 = 33926231.46 (interval #89) if we assume k = pi/log(3/2) in
%  the entry 4378640k - 2/5 (which is 33926231.43)

   \bibitem{Steuding}
   \textsc{J.~Steuding},	% J\"orn Steuding
   \emph{Value-distribution of L-functions},
   Lecture Notes in Mathematics {\bf{1877}}, 
   Springer, 2007.
    
   \bibitem{T}
   \textsc{E.~C.~Titchmarsh},
    \emph{The Theory of the Riemann Zeta-function},
   2nd edition, edited by D.\ R.\ Heath-Brown. 
   The Clarendon Press, Oxford, 1986.

   \bibitem{Tsang}
   \textsc{K.~Tsang},	% Kai-man Tsang
   \emph{The Distribution of the Values of the Riemann Zeta-function},
   PhD Thesis, Department of Mathematics, Princeton University, Oct.\ 1984.
   % http://www.math.sjsu.edu/~goldston/Tsang[1-3].pdf for Chapter 1
   % See AMS Notices, 56, 6 (June/July 2009), 692-710 for Tsang's comments
   % on being Selberg's student (the only one?).

   \bibitem{Wrench}
   \textsc{J.~W.~Wrench}, 
   \emph{Evaluation of Artin's constant and the twin prime constant},
   Math.\ Comp.\ {\bf{15}} (1961), 396--398.

   \end{thebibliography}