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\leaf{Distribution of the zeros of the Riemann Zeta function}

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One of the most celebrated problem of mathematics is the {\em Riemann
  hypothesis} which states that all the non trivial zeros of the
  Zeta-function lie on the critical line $\Re(s)=1/2$.
Even if this famous problem is unsolved for so long, a lot of things
are known about the zeros of $\zeta(s)$ and we expose here
the most classical related results~: all the non trivial zeros lie in
  the critical strip, the number of such zeros with ordinate less than
  $T$ is proportional to $T\log T$, most zeros concentrate along the
  critical line $\sigma=1/2$, there exists an infinity of zeros on the
  critical line and moreover, more than two fifth of the zeros are on
  the critical line.

\section{Generalities}

Let us recall (see section on the analytic continuation of $\zeta(s)$
in~\href{zetageneralities.html}{The Riemann Zeta-function :
  generalities}) that zeta vanishes  
at negative odd integers. These zeros are called the
{\em trivial zeros} of $\zeta(s)$. The functional equation
\begin{equation}\label{eq.FunctionalEq}
\zeta(s) = \chi(s)\zeta(1-s),\qquad \chi(s) = 2^s \pi^{s-1} \sin 
\left( \frac{\pi s}{2} \right) \Gamma(1-s).
\end{equation}
entails that other zeros of $\zeta(s)$ (they are called the {\em non
  trivial zeros}) are symetric with respect to the critical line
  $\Re(s)=1/2$~: for each non trivial zero $s=\sigma+it$, the value
  $s'=1-\sigma+it$ is also a zero of $\zeta(s)$.

\subsection*{The non trivial zeros lie in the critical strip}

We show  that all the non trivial zeros of $\zeta(s)$
lie in the {\em critical strip} defined by values of the complex
number $s$ such that $0<\Re(s)<1$. Because of the functional equation,
it suffices to show that $\zeta(s)$ does not vanish on the closed half
plane $\Re(s) \geq 1$. 

The Euler infinite product (see~\href{zetageneralities.html}{The Riemann
  Zeta-function : generalities}) 
$$\zeta(s) = \prod_{p\ \text{prime}} \frac{1}{1-p^{-s}},$$
valid for all complex numbers $s$ with $\Re(s)>1$, shows that
$\zeta(s)$ does not vanish for $\Re(s)>1$ (a convergent infinite
product can not converge to zero because its logarithm is a convergent
series). Thus it suffices now to prove that $\zeta(s)$ does not vanish
on the line $\Re(s)=1$. This property is in fact the key in the proof
of the prime number theorem, and Hadamard and De La Vall\'ee
Poussin obtained this result independantly in 1896 by different mean (this
problem is in fact a first step in a determination of a zero-free
region, important to obtain good error terms in the prime number theorem).
We present here the argument of De La Vall\'ee Poussin which is 
simpler to expose and more elegant, in a form close to the
presentation of~\cite{Titchmarsh}.

\subsubsection*{The Zeta-function has no zeros on the line $\Re(s)=1$}

The starting point is the relation
\begin{equation}
  \label{eq:DeLaValleePoussinTrick}
3+4\cos\phi + \cos 2\phi = 2(1+\cos\phi)^2 \geq 0  
\end{equation}
for all values of the real number $\phi$. The Euler infinite product
writes as
$$\zeta(s) = \exp \left( \sum_{p \ \text{prime}} \sum_{m=1}^{\infty}
  \frac{1}{mp^{ms}} \right),\qquad \Re(s)>1$$
thus for the complex number $s=\sigma+it$ we have
$$|\zeta(s)| = \exp \left( \sum_{p \ \text{prime}} \sum_{m=1}^{\infty}
  \frac{\cos(mt\log p)}{mp^{m\sigma}} \right),\qquad
  \sigma=\Re(s)>1.$$
This relation entails
\begin{eqnarray*}
\lefteqn{\zeta(\sigma)^3 |\zeta(\sigma+it)|^4 |\zeta(\sigma +2it)|} \\
& & =\exp \left( \sum_{p \ \text{prime}} \sum_{m=1}^{\infty}
\frac{3+4\cos(mt\log p)+\cos(2mt\log p)}{mp^{m\sigma}}\right)
\end{eqnarray*}
so with~(\ref{eq:DeLaValleePoussinTrick}) we deduce
\begin{equation}
  \label{eq:DeLaValleePoussin2}
\zeta(\sigma)^3 |\zeta(\sigma+it)|^4 |\zeta(\sigma +2it)| \geq
1,\qquad \sigma>1.  
\end{equation}
Now suppose that $1+it$ is a zero of $\zeta(s)$. Letting $\sigma \to
1$, we have $\zeta(\sigma) \sim 1/(\sigma-1)$ and
$\zeta(\sigma+it)=O(\sigma-1)$, so that
$\zeta(\sigma)^3 |\zeta(\sigma+it)|^4 = O(\sigma-1)$ thus as $\sigma
\to 1$,~(\ref{eq:DeLaValleePoussin2}) entails that
$|\zeta(\sigma +2it)|$ tends to infinity, which is impossible since
$\zeta(s)$ is analytic around $1+2it$. Thus we have proved that
$\zeta(s)$ has no zeros on the line $\Re(s)=1$.

Other proofs of this result can be found in~\cite{Titchmarsh}.

\section{Counting the number of zeros in the critical strip}

Even if precise location of zeros are not known, a lot of results have
been obtained on counting the number of zeros in the critical strip.
All the results in this section are more proved and detailed
in~\cite[Ch. 9]{Titchmarsh}.

\subsection{The number of zeros less than a given height}

We let
\begin{eqnarray*}
\theta(t) & = & \arg \left( \pi^{-it/2} \Gamma( 1/4 + it/2 ) \right) \\
S(t) & = & \frac{1}{\pi} \arg \zeta( 1/2+it) \\
\end{eqnarray*}
where the arguments are defined by continuous variation
of $s$ starting with the value $0$ at $s=2$, going up vertically to
$s=2+it$ and then horizontally to $s=\frac{1}{2}+it$ (when there is a
zero of Zeta on the segment between $\frac{1}{2}+it$ and $2+it$,
$S(T)$ is defined by $S(T+0)$).
We already encountered the $\theta(t)$ function
in~\href{zetaevaluations.html#eq:theta2}{Numerical 
  evaluation of the Riemann Zeta-function} while defining the
Riemann-Siegel function $Z(t)$.
If $N(T)$ denotes the number of zeros of $\zeta(\sigma+it)$ in
the region $0<\sigma<1$, $0<t\leq T$, then a result
in\cite{Titchmarsh} states that
$$
N(T) = 1 + \frac{\theta(T)}{\pi} + S(T),
$$
and using the asymptotic expansion of $\theta(T)$
(see~\href{zetaevaluations.html#eq:thetaAsymptotic}{Numerical  
  evaluation of the Riemann Zeta-function}) we have
\begin{equation}
  \label{eq:N(t)}
N(T) = \frac{T}{2\pi} \log \frac{T}{2\pi} - \frac{T}{2\pi}
+ \frac{7}{8} +  S(T) + O\left( \frac{1}{T} \right).
\end{equation}
The function $S(T)$ satisfies
\begin{equation}
  \label{eq:Sbound}
  S(T) = O(\log T),\qquad T\to +\infty,
\end{equation}
which permits to say that $N(T)$ behaves like 
$T/(2\pi) \log(T/(2\pi))$. 
So the zeros of the Zeta-function become more and
more dense as one goes up in the critical strip. More precisely
the height of the $n$-th zero (ordered in increasing values of its
ordinate) behaves like $2\pi n/\log n$. The results above also permit
to state that the gap between the ordinates of successive zeros is bounded.

\subsubsection*{More on the function $S(T)$}

The function $S(T)$ is important in local study of zeros and appears
to be quite complicated. Today, no improvement of the
bound~(\ref{eq:Sbound}) is known (under the Riemann  hypothesis, we
have the strongest bound $S(T) = O(\log T/\log\log T)$), but other
results have been obtained. For example, we have the bound
\begin{equation}
  \label{eq:S1bound}
  \int_0^T S(t) dt = O(\log T),
\end{equation}
which in particular, entails that the average value of $S(T)$ is
zero. It is also known that $S(T)$ is not too small, and more
precisely, a result from Selberg states that there exists a constant
$A>0$ for which the inequality
$$|S(T)| > A (\log T)^{1/3}(\log\log T)^{-7/3}$$
holds for an infinity of values of $T$ tending to infinity. 
Another result from Selberg gives
$$\int_0^T S(t)^2\,dt \sim \frac{1}{2\pi^2} T \log\log T,$$
thus the average value of $S(t)^2$ on $[0,T]$ is $1/(2\pi^2) \log\log
T$ (this result has latter been refined by Ghosh in 1983 who proved
that $|S(T)|/(\log\log T)^{1/2}$ has a limiting distribution).
So $S(t)$ is, in average, tending to infinity very very slowly.

Finally a result of Titchmarsh states that the function $S(T)$
changes its sign infinitly often.

\subsection{The zeros off the critical line}

Even if Riemann hypothesis has not been proved, it is known that the
zeros of the Zeta-function concentrate along the critical line. To
precise this result, we define, when $0<\sigma<1/2$, the function
$N(\sigma,T)$ as the number of zeros $s=\beta+it$ of Zeta such that
$\beta>\alpha$ and $0< t \leq T$ (under the Riemann hypothesis one
has $N(1/2,T)=0$).
A first result about $N(\sigma,T)$ states that for any fixed
$\sigma>1/2$, one has
\begin{equation}
  \label{eq:NsigmaBound}
N(\sigma,T) = O(T).  
\end{equation}
Since $N(T) \sim T/(2\pi) \log(T/(2\pi))$, we deduce that for any
$\delta>0$, all but an infinitesimal proportion of the zeros lie in
the strip $1/2-\delta < \sigma < 1/2+\delta$.

There exists numerous results that improve the
bound~(\ref{eq:NsigmaBound}) in some context, for example we have
\begin{eqnarray*}
N(\sigma,T) & = & O(T^{4\sigma(1-\sigma)+\epsilon}),\quad \text{for
  all } \ \epsilon>0, \\
N(\sigma,T) & = & O(T^{3/2-\sigma} \log^5 T) \\
N(\sigma,T) & = & O(T^{3(1-\sigma)/(2-\sigma)} \log^5 T).
\end{eqnarray*}

A result of a different kind has been obtained by Selberg, who
proved
$$\int_{1/2}^1 N(\sigma,T) d\sigma = O(T).$$

\section{The zeros on the critical line}

The Riemann hypothesis states that all non trivial zeros of $\zeta(s)$
lie on the critical line $\Re(s)=1/2$. Even if this has never been
proved or disproved, mathematicians succeeded in proving that there
exists an infinity of non trivial zeros on the critical line, and
later, that a positive proportion of the zeros are on the line.

\subsection{An infinity of zeros are on the critical line}

Hardy was the first to prove in 1914 that an infinity of zeros are on the
critical line. Later, other mathematicians like P\'olya in 1927, Landau, or
Titchmarsh in the 1930's gave other proofs. In~\cite{Titchmarsh}, 
five different proofs are given.

We sketch here the ideas under the Titchmarsh approach, which makes use
the Riemann-Siegel $Z$ function (see
\href{zetaevaluations.html#eq:HardyZ}{The Riemann-Siegel Z-function}
in {\em Numerical evaluation of the Riemann Zeta-function}), with
considerations close to what is used in numerical computations of the
zeros. We recall that the $Z(t)$ function is purely real and satisfies
$|Z(t)|=|\zeta(1/2+it)|$, and that Riemann-Siegel formula for $Z(t)$
writes as
\begin{equation}
  \label{eq:ZRiemannSiegel2}
Z(t) = 2 \sum_{1 \leq n \leq x} \frac{\cos(\theta(t)-t\log
  n)}{\sqrt{n}} + O(t^{-1/4}),\qquad x=\sqrt{\frac{t}{2\pi}}.  
\end{equation}
Titchmarsh observed that the dominant term in the sum above is
obtained with $n=1$ and is $\cos \theta(t)$ and thus one should
expect that on average, the sign of $Z(t)$ would be the sign of
$\cos \theta(t)$. For that reason he defined $t_{\nu}$ as the
solution of $\theta(t_{\nu})=\nu \pi$ (we have $t_{\nu} \sim 2\pi
\nu/\log \nu$) and showed that on average, the value of $Z(t_{\nu})$
is $2 (-1)^{\nu}$. More precisely, he proved that
\begin{equation}
  \label{eq:sumZnu}
\sum_{\nu \leq N} Z(t_{2\nu}) \sim 2N,\qquad
\sum_{\nu \leq N} Z(t_{2\nu+1}) \sim -2N  
\end{equation}
by showing that in the $\nu$ summation, the  terms 
$\cos(\theta(t_{2\nu})-t_{2\nu}\log n) n^{-1/2}$ and
$\cos(\theta(t_{2\nu+1})-t_{2\nu+1}\log n) n^{-1/2}$
of~(\ref{eq:ZRiemannSiegel2}) for $n \geq 2$
cancellate and give a contribution  of inferior order.
It is now easy to prove that $Z(t)$ has an infinity
of zeros since if not, it would keep the same sign for $t$ large
enough, thus one of the two estimates in~(\ref{eq:sumZnu}) would not
be satisfied.

\subsection{A finite proportion of zeros lie on the critical line}

After Hardy proof that there exists an infinity of zeros on the
critical line, strongest results that estimate the minimal number of
zeros on the line have been obtained. Denoting by $N_0(T)$ the number
of zeros of $\zeta(1/2+it)$ for $0<t\leq T$, the first strongest
historical result is due to Hardy and Littlewood who proved in 1921
that there exists a constant $A>0$ for which the inequality 
$$N_0(T) > AT$$
holds for all values of $T$ (see~\cite{Titchmarsh} for a
proof). Selberg in 1932 improved considerably this result by showing
the existence of a constant $A>0$ for which 
$$N_0(T) > A T\log T$$
holds (the numerical value of $A$ is Selberg proof is very
small). Since $N(T) \sim T/(2\pi) \log T$, it means that a finite
proportion of the zeros lie on the critical line. The proof is quite
complicated and is given in~\cite{Titchmarsh}.
The most significant result on $N_0(T)$ has been obtained by Conrey 
in 1989 (see~\cite{Conrey}) and states that for $T$ large enough, one as
$$N_0(T) \geq \alpha N(T),\qquad \alpha=0.40219.$$
Thus more than one third of the zeros lie on the critical line.

\bibliographystyle{plain}

\begin{thebibliography}{3}

\bibitem{Conrey} J. B. Conrey, ``More than two fifths of the zeros of
  the Riemann zeta function are on the critical line'', {\em J. reine
  angew. Math.}, 399 (1989), pp. 1-26.

\bibitem{Edwards} H. M. Edwards, {\em Riemann's Zeta Function},
Academic Press, 1974.

\bibitem{Titchmarsh}  E. C. Titchmarsh, 
{\em The theory of the Riemann Zeta-function}, Oxford Science publications,
second edition, revised by D. R. Heath-Brown (1986).



\end{thebibliography}

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