text:
\section{Introduction}

The distribution of zeros of Riemann's zeta function is one of the central
problems in modern mathematics. The most famous conjecture is that all of
the zeros are on the critical line $\mathrm{Re}s=1/2.$ A more precise
version says that the zeros behave like eigenvalues of large Hermitian
matrices. While these statements are conjectural, a great deal is known
about the statistical properties of these zeros and about zeros of closely
related functions. In this report we aim to summarize findings in this
research area.

We give necessary background information, and we cover the three main types
of zeta functions: number-theoretical, Selberg-type, and dynamical zeta
functions.

 Some interesting and important topics are left outside of the
scope of this report. For example, we do not discuss quantum arithmetic
chaos or characteristic polynomials of random matrices.

The paper is divided in three main sections according to the type of the
zeta function we discuss. Inside each section we tried to separate the
discussion of the properties of zeros at the global and local scales.

Let us briefly describe these types of zeta functions and their
relationships. First, the number-theoretical zeta functions come from
integers in number fields and their generalizations. Due to the additive and
multiplicative structures of the integers, and in particular due to the
unique decomposition in prime factors, the zeta functions have a functional
equation, $\zeta \left( 1-s\right) =c\left( s\right) \zeta \left( s\right) $
and the Euler product formula $\zeta \left( s\right) =\prod_{\mathfrak{p
}\left( 1-\left( N\mathfrak{p}\right) ^{s}\right) ^{-1}.$

One can also look at number-theoretical zeta functions from a different
prospective if one starts with modular forms, that is, functions on two
dimensional lattices that are invariant relative to transformations from
SL_{2}\left( Z\right) .$

Some modular forms (the cusp forms) can be written as
\sum_{n>0}c_{n}\exp \left( 2\pi inz\right) ,$ where $z$ is the ratio of the
periods of the lattice, and one can associate a zeta function $\sum
c_{n}n^{-s}$ to this modular form. Then the modularity ensures that the zeta
function satisfies a functional equation. In addition, the zeta function will have
the Euler product property if the original modular form is an eigenvector
for certain operators (the Hecke operators).

This class of modular zeta functions overlaps with number-theoretical
functions to a large extent.

In fact, a recent significant development
occurred when it was proved that all zeta functions associated with elliptic
curves come from modular zeta functions. Among other applications, this
discovery was a key to the proof of Fermat's last theorem.

An important representative of the second class of zetas is Selberg's zeta function.  Let $\mathbb{H}$ is the upper
half-plane with the hyperbolic metric and $\Gamma $ is a discrete subgroup
of $SL_{2}\left( \mathbb{Z}\right) $.
Selberg showed that sums over eigenvalues of the Laplace operator on Riemann
surface $\Gamma \backslash \mathbb{H}$ are related to sums over
non-conjugate elements of $\Gamma .$ This relation is called Selberg's
trace formula. Selberg's zeta function is constructed in such a way that it
has the same relation to the trace formula as Riemann's zeta function has to
the so-called \textquotedblleft explicit formula\textquotedblright\ for sums
over zeros of the zeta function.

While Selberg's zeta function resembles Riemann's zeta in some features,
there are significant differences. In particular, the statistical behavior
of its zeros depends on the group $\Gamma $ and often it is significantly
different from the behavior of Riemann's zeros.

The third class of zetas, the dynamical zeta functions, appeared first as a generalization of Weil's zeta
functions.

Let $k_{q}$ is a finite field with $q$ elements and $f\left(
X,Y\right) $ is an absolutely irreducible polynomial over this field. The simplest Weil's zeta function is the usual number-theoretical
zeta functions for integer ideals of the field $k_{q}\left[ X,Y\right] /f\left(
X,Y\right) .$ It turns out
(non-trivially) that this zeta function can be written as a generating
function for fixed points of powers of the Frobenius map: $x\rightarrow
x^{q}.$ (The Frobenius map acts on the points of the curve $f\left(
X,Y\right) $ in the algebraic closure of $k_{q}.$)

The theory of dynamical zeta functions starts from this setup and defines a
zeta function for arbitrary map $F$ on a set $M$ as the
generating function for fixed points of powers of $F.$ This definition can
also be generalized to flows on a set, that is, to maps $F:M\times \mathbb{R
^{+}\rightarrow M.$ In particular, Selberg's zeta function can be understood as the dynamical zeta
function for the geodesic flow on the surface $\Gamma \backslash \mathbb{H}$.

With this overview in mind we now come to a more detailed description of
available results about the statistics of zeta function zeros.

\section{Number-Theoretical Zetas}

\subsection{Riemann's zeta}

There are several excellent sources on Riemann's zeta and Dirichlet
L-functions, for example the books by Davenport \cite{davenport67} and
Titchmarsh \cite{titchmarsh86}. In addition, a very good reference for all topics in this report is provided by Iwaniec and Kowalski's book \cit
{iwaniec_kowalski04}.

By definition Riemann's zeta function is given by the serie
\begin{equation}
\zeta \left( s\right) =\sum_{n=1}^{\infty }\frac{1}{n^{s}},
\label{definition_Riemann_zeta}
\end{equation
for $\mathrm{Re}s>1.$ It can be analytically continued to a meromorphic
function in the entire complex plane and it satisfies the functional equation
\begin{equation}
\zeta \left( 1-s\right) =\pi ^{s-\frac{1}{2}}\frac{\Gamma \left( \frac{1}{2
\left( 1-s\right) \right) }{\Gamma \left( \frac{1}{2}s\right) }\zeta \left(
s\right) .  \label{functional_equation_Riemann}
\end{equation
Indeed, we can relate  $\zeta \left( s\right) $ to the series $\theta \left( x\right)
=\sum_{n=1}^{\infty }e^{-n^{2}\pi x},$
\begin{equation*}
\frac{\Gamma \left( \frac{1}{2}s\right) \zeta \left( s\right) }{\pi ^{s/2}
=\sum_{n=1}^{\infty }\int_{0}^{\infty }x^{s/2-1}e^{-n^{2}\pi
x}dx=\int_{0}^{\infty }x^{s/2-1} \theta \left( x\right) dx.
\end{equation*
 From the identities for the Jacobi
theta-functions, implied by the Poisson summation formula, it follows that
\begin{equation*}
2\theta \left( x\right) +1=\frac{1}{\sqrt{x}}\left( 2\theta \left( \frac{1}
\sqrt{x}}\right) +1\right) .
\end{equation*
Writing
\begin{equation*}
\int_{0}^{\infty }x^{s/2-1}\theta \left( x\right)
dx=\int_{0}^{1}x^{s/2-1}\theta \left( x\right) dx+\int_{1}^{\infty
}x^{s/2-1}\theta \left( x\right) dx,
\end{equation*
and applying the identity to the integral from $0$ to $1,$ we find that
\begin{equation*}
\frac{\Gamma \left( \frac{1}{2}s\right) \zeta \left( s\right) }{\pi ^{s/2}}
\frac{1}{s\left( s-1\right) }+\int_{1}^{\infty }\left(
x^{-s/2-1/2}+x^{s/2-1}\right) \theta \left( x\right) dx,
\end{equation*
which is symmetric relative to the change $s\rightarrow 1-s.$

The second fundamental property of Riemann's zeta is the Euler product
formula:
\begin{equation}
\zeta \left( s\right) =\prod_{p}\left( 1-\frac{1}{p^{s}}\right) ^{-1}
\label{product_formula_Riemann}
\end{equation
valid for $\mathrm{Re}s>1.$

Let the non-trivial zeros of Riemann's zeta function be
denoted by $\rho _{k}=\beta _{k}+i\gamma _{k}.$ The Riemann's hypothesis
asserts that $\beta _{k}=1/2$ and it is known that $0<\beta _{k}<1.$ The
zeros are symmetric relative to the real axis, so it is enough to consider
zeros with positive imaginary part, $\gamma _{k}>0$. We order them so that
the imaginary part is non-decreasing, $\gamma _{1}\leq \gamma _{2}\leq
\ldots .$

One of the most important ideas in the field is to relate the sums over prime numbers and sums over zeta function
zeros. For example, the Riemann-Mangoldt formula says that

\begin{equation}
\sum_{n\leq x}\Lambda \left( n\right) =x-\sum_{\rho }\frac{x^{\rho }}{\rho
+\sum_{n}\frac{x^{-2n}}{2n}-\frac{\zeta ^{\prime }}{\zeta }\left( 0\right) ,
\label{Riemann_Mangoldt}
\end{equation}
where $\Lambda \left( n\right) =\log p,$ if $n$ is a prime $p$ or a power of
$p,$ and otherwise $\Lambda \left( n\right) =0$.
The idea of the proof (due to Riemann) is to consider the logarithmic derivative
\begin{equation}
\frac{\zeta ^{\prime }}{\zeta }\left( s\right) =-\sum_{n=2}^{\infty }\frac
\Lambda \left( n\right) }{n^{s}},  \label{log_derivative_Riemann}
\end{equation
 and then to use the following formula:

\begin{equation*}
\frac{1}{2\pi i}\int_{c-i\infty }^{c+i\infty }y^{s}\frac{ds}{s}=\left\{
\begin{array}{cc}
0 & \text{if }0<y<1, \\
\frac{1}{2} & \text{if }y=1, \\
1 & \text{if }y>1
\end{array
\right.
\end{equation*
where $c>0$, in order to pick out the terms in a Dirichlet series with $n\leq x$ by
taking $y=x/n.$

From (\ref{log_derivative_Riemann}), one gets
\begin{equation*}
\sum_{n\leq x}\Lambda \left( n\right) =\frac{1}{2\pi i}\int_{c-i\infty
}^{c+i\infty }\left[ -\frac{\zeta ^{\prime }}{\zeta }\left( s\right) \right]
x^{s}\frac{ds}{s}.
\end{equation*
Moving the line of integration away to infinity on the left and collecting
the residues at the poles one finds formula (\ref{Riemann_Mangoldt}). (See
Chapter 17 in Davenport \cite{davenport67} for a detailed proof.)

More generally, the Landau's formula holds:
\begin{equation}
\sum_{n\leq x}\frac{\Lambda \left( n\right) }{n^{s}}=\frac{x^{1-s}}{1-s
-\sum_{\rho }\frac{x^{\rho -s}}{\rho -s}+\sum_{n}\frac{x^{-2n-s}}{2n+s}
\frac{\zeta ^{\prime }}{\zeta }\left( s\right) .  \label{Landau}
\end{equation}

Formula (\ref{Riemann_Mangoldt}) is useful if one want to study the
distribution of primes and something is known about the distribution of
zeros, and the formula (\ref{Landau}) can be used to study the behavior of
\frac{\zeta ^{\prime }}{\zeta }\left( s\right) $ if some information about
the primes is known.

One difficulty is that the sums over zeros in these formulas are only
conditionally convergent. ($\rho _{k}\approx 1/2+2\pi i\frac{k}{\log k}$).
In particular, in these sums it is assumed that the summation is in the
order of increasing $\left\vert \mathrm{Im}\rho \right\vert $.

Another variant of this formula was discovered by Selberg and it avoids the
problem of conditional convergence. Define
\begin{equation*}
\Lambda _{x}\left( n\right) =\left\{
\begin{array}{cc}
\Lambda \left( n\right) , & \text{ for }1\leq n\leq x, \\
\Lambda \left( n\right) \left( \frac{\log ^{2}\frac{x^{3}}{n}-2\log ^{2
\frac{x^{2}}{n}}{2\log ^{2}x}\right) , & \text{for }x\leq n\leq x^{2}, \\
\Lambda \left( n\right) \frac{\log ^{2}\frac{x^{3}}{n}}{2\log ^{2}x}, &
\text{for }x^{2}\leq n\leq x^{3}
\end{array
\right.
\end{equation*
Then,
\begin{eqnarray}
\frac{\zeta ^{\prime }}{\zeta }\left( s\right) &=&-\sum_{n\leq x^{3}}\frac
\Lambda _{x}\left( n\right) }{n^{s}}+\frac{1}{\log ^{2}x}\sum_{\rho }\frac
x^{\rho -s}\left( 1-x^{\rho -s}\right) ^{2}}{\left( s-\rho \right) ^{3}}
\label{explicit_formula_Selberg} \\
&&+\frac{x^{1-s}\left( 1-x^{1-s}\right) ^{2}}{\log ^{2}x\cdot \left(
1-s\right) ^{3}}+\frac{1}{\log ^{2}x}\sum_{q=1}^{\infty }\frac
x^{-2q-s}\left( 1-x^{-2q-s}\right) ^{2}}{\left( 2q+s\right) ^{3}}.  \notag
\end{eqnarray}

Finally, there is one more variant of the explicit formula (sometimes called
Delsarte's explicit formula). Suppose that $H\left( s\right) $ is an
analytic function in the strip $-c\leq \mathrm{Im}$ $s\leq 1+c$ (for $c>0$)
and that $\left\vert H\left( \sigma +it\right) \right\vert \leq A\left(
1+\left\vert t\right\vert \right) ^{-\left( 1+\delta \right) }$ uniformly in
$\sigma $ in the strip. Let $h\left( t\right) =H\left( \frac{1}{2}+it\right)
$ and define
\begin{equation*}
\widehat{h}\left( x\right) =\int_{\mathbb{R}}h\left( t\right) e^{-2\pi
itx}dt.
\end{equation*
(Note that analyticity of $H\left( s\right) $ implies that $\widehat{h
\left( x\right) $ has finite support.) Then,
\begin{eqnarray}
\sum_{\rho }H\left( \rho \right) &=&H\left( 0\right) +H\left( 1\right)
\frac{1}{2\pi }\sum_{n=1}^{\infty }\frac{\Lambda \left( n\right) }{\sqrt{n}
\left[ \widehat{h}\left( \frac{\log n}{2\pi }\right) +\widehat{h}\left(
\frac{\log n}{2\pi }\right) \right]  \label{formula_delsarte} \\
&&-\frac{1}{2\pi }\int_{-\infty }^{\infty }h\left( t\right) \Psi \left(
t\right) dt,  \notag
\end{eqnarray
where
\begin{equation*}
\Psi \left( t\right) =\frac{\Gamma ^{\prime }}{\Gamma }\left( \frac{1}{2
+it\right) +\frac{\Gamma ^{\prime }}{\Gamma }\left( \frac{1}{2}-it\right) .
\end{equation*}

The idea of the proof is similar to the proof of the Riemann-Mangoldt
formula. One starts with the formula
\begin{eqnarray*}
\frac{1}{2\pi i}\int_{2-i\infty }^{2+i\infty }\left[ -\frac{\zeta ^{\prime
}{\zeta }\left( s\right) \right] H\left( s\right) ds &=&-\sum_{n=1}^{\infty
\frac{\Lambda \left( n\right) }{n^{1/2}}\frac{1}{2\pi }\int_{-\infty
}^{\infty }h\left( t\right) e^{-i\left( \log n\right) t}dt \\
&=&-\frac{1}{2\pi }\sum_{n=1}^{\infty }\frac{\Lambda \left( n\right) }
n^{1/2}}\widehat{h}\left( \frac{\log n}{2\pi }\right) .
\end{eqnarray*}

Next, one can move the line of integration to $\left( -1-i\infty ,-1+i\infty
\right) $ and use the calculus of residues and the functional equation to
obtain formula (\ref{formula_delsarte}).

The most general explicit formula was derived by Weil (see \cite{weil52}).
We will not present it here since it involves adelic language and this would
take us too far apart.

\subsubsection{Statistics of zeros on global scale}

Let $\mathcal{N}(T)$ denote the number of zeros with the imaginary part
strictly between $0$ and $T$. If there is a zero with imaginary part equal
to $T$, then we count this zero as $1/2$. Define
\begin{equation*}
S(T):=\frac{1}{\pi }\mathrm{Im}\log \zeta (\frac{1}{2}+iT),
\end{equation*
where the logarithm is calculated by continuous variation along the contour
\sigma +iT,$ with $\sigma $ changing from $+\infty $ to $1/2.$

By applying the argument principle to $\zeta $ and utilizing the functional
equation, it is possible to show (see Chapter 15 in \cite{davenport67}) that
\begin{equation*}
\mathcal{N}(T)=\frac{T}{2\pi }\log \frac{T}{2\pi e}+\frac{7}{8}+S\left(
T\right) +O\left( \frac{1}{1+T}\right) .
\end{equation*
Let
\begin{equation*}
X\left( t\right) :=\frac{\sqrt{2}\pi S(t)}{\sqrt{\log \log t}}.
\end{equation*
Then, we have the following theorem by Selberg. (See \cite{selberg44} and
\cite{selberg46a}.)

\begin{theorem}[Selberg]
\label{theorem_selberg} Assume RH, and let $T^{a}\leq H\leq T^{2},$ where
a>0.$ Then for every $k\geq 1$ \
\begin{equation*}
\frac{1}{H}\int_{T}^{T+H}\left| X\left( t\right) \right| ^{2k}dt=\frac{2k!}
k!2^{k}}+O(1/\log \log T),
\end{equation*
with the constant in the remainder term that depends only on $k$ and $a.$
\end{theorem}

In other words, $X\left( t\right) $ behaves like a Gaussian random variable.
Note that the Riemann Hypothesis is not assumed in this result.

Recently, this was generalized by Bourgade in \cite{bourgade10}, who found
the correlation structure of $X\left( t\right) .$

\begin{theorem}[Bourgade]
\label{theorem_bourgade} Let $\omega $ be uniform on $(0,1)$, $\epsilon
_{t}\rightarrow 0$, $\epsilon _{t}\gg 1/\log t$, and functions $0\leq
f_{t}^{\left( 1\right) }<\ldots <f_{t}^{\left( l\right) }<c<\infty .$
Suppose that for all $i\neq j,
\begin{equation*}
\frac{\log \left| f_{t}^{\left( j\right) }-f_{t}^{\left( i\right) }\right| }
\log \epsilon _{t}}\rightarrow c_{i,j}\in \left[ 0,\infty \right] .
\end{equation*
Then the vector
\begin{equation*}
\frac{1}{\sqrt{-\log \epsilon _{t}}}\left( \log \zeta \left( \frac{1}{2
+\varepsilon _{t}+if_{t}^{\left( 1\right) }+i\omega t,\ldots ,\frac{1}{2
+\varepsilon _{t}+if_{t}^{\left( l\right) }+i\omega t\right) \right)
\end{equation*
converges in law to a complex Gaussian vector $\left( Y_{1},\ldots
,Y_{l}\right) $ with the zero mean and covariance function
\begin{equation*}
\mathrm{Cov}\left( Y_{i},Y_{j}\right) =\left\{
\begin{array}{cc}
1 & \text{if }i=j, \\
1\wedge c_{i,j} & \text{if }i\neq j
\end{array
\right.
\end{equation*
Moreover, the above result remains true if $\epsilon _{t}\ll 1/\log t$,
replacing the normalization $-\log \varepsilon _{t}$ with $\log \log t.$
\end{theorem}

\bigskip

This result implies the following Corollary for the zeros of the Riemann's
zeta. Let
\begin{equation*}
\Delta \left( t_{1},t_{2}\right) =\left( \mathcal{N}(t_{2})-\frac{t_{2}}
2\pi }\log \frac{t_{2}}{2\pi e}\right) -\left( \mathcal{N}(t_{1})-\frac{t_{1
}{2\pi }\log \frac{t_{1}}{2\pi e}\right) ,
\end{equation*
which represents the number of zeros with imaginary part between $t_{1}$ and
$t_{2}$ minus its deterministic prediction.

\begin{corollary}[Bourgade]
\label{corollary_bourgade} Let $K_{t}$ be such that, for some $\varepsilon
>0 $ and all $t$, $K_{t}>\varepsilon $. Suppose $\log K_{t}/\log \log
t\rightarrow \delta \in \lbrack 0,1),$ as $t\rightarrow \infty .$ Then the
finite-dimensional distributions of the proces
\begin{equation*}
\frac{\Delta \left( \omega t+\alpha /K_{t},\omega t+\beta /K_{t}\right) }
\frac{1}{\pi }\sqrt{\left( 1-\delta \right) \log \log t}},\text{ }0\leq
\alpha <\beta <\infty ,
\end{equation*
converge to those of a centered Gaussian process ($\widetilde{\Delta }\left(
\alpha ,\beta \right) $, $0\leq \alpha <\beta <\infty $) with the covariance
structur
\begin{equation*}
\mathbb{E}\left( \widetilde{\Delta }\left( \alpha ,\beta \right) \widetilde
\Delta }\left( \alpha ^{\prime },\beta ^{\prime }\right) \right) =\left\{
\begin{array}{cc}
1 & \text{if }\alpha =\alpha ^{\prime },\text{ and }\beta =\beta ^{\prime },
\\
1/2 & \text{if }\alpha =\alpha ^{\prime },\text{ and }\beta \neq \beta
^{\prime }, \\
1/2 & \text{if }\alpha \neq \alpha ^{\prime },\text{ and }\beta =\beta
^{\prime }, \\
-1/2 & \text{if }\beta =\alpha ^{\prime }, \\
0 & \text{elsewhere.
\end{array
\right.
\end{equation*}
\end{corollary}

Note that $K_{t}$ is of order $\left( \log t\right) ^{-\delta }$ and the
average spacing between zeros is $1/\log t,$ hence the number of zeros in
the interval $\left( \omega t+\alpha /K_{t},\omega t+\beta /K_{t}\right) $
is of order $\left( \log t\right) ^{1-\delta }.$

This result perfectly corresponds to the results of Diaconis and Evans in
\cite{diaconis_evans01} about fluctuations of eigenvalues of random unitary
matrices.

Selberg and Bourgade's theorems are based on an approximation the functions
S\left( t\right) $, discovered by Selberg.

\begin{proposition}[Selberg]
\label{proposition_approximation}Suppose $k\in \mathbb{Z}^{+},$ $0<a<1.$
Then there exists $c_{a,k}>0$ such that for any $1/2\leq \sigma \leq 1$ and
t^{a/k}\leq x\leq t^{1/k},$ it is true that
\begin{equation*}
\frac{1}{t}\int_{1}^{t}\left\vert \log \zeta \left( \sigma +is\right)
-\sum_{p\leq x^{3}}\frac{1}{p^{\sigma +is}}\right\vert ^{2k}ds\leq c_{a,k}.
\end{equation*}
\end{proposition}

The proof of this statement, in turn, depends on Selberg's formula (\re
{explicit_formula_Selberg}).

\subsubsection{Statistics of zeros on local scale}

At the local scale, we have a result due to Montgomery, \cite{montgomery73}.
Assume the Riemann Hypothesis, and define
\begin{equation*}
F\left( \alpha \right) =F\left( \alpha ,T\right) =\frac{2\pi }{T\log T
\sum_{0<\gamma ,\gamma ^{\prime }<T}T^{i\alpha \left( \gamma -\gamma
^{\prime }\right) }w\left( \gamma -\gamma ^{\prime }\right) ,
\end{equation*
where $\alpha $ is real, $T\geq 2,$ and $w\left( u\right) $ is a weighting
function,
\begin{equation*}
w(u)=\frac{4}{4+u^{2}}.
\end{equation*}

\begin{theorem}[Montgomery]
\label{theorem_Montgomery} The function $F\left( \alpha \right) $ is real
and $F\left( \alpha \right) =F\left( -\alpha \right) .$ If $T>T_{0}\left(
\varepsilon \right) ,$ then $F\left( \alpha \right) \geq -\varepsilon $ for
all $\alpha .$ For fixed $\alpha $ satisfying $0\leq \alpha <1$ we have
\begin{equation*}
F\left( \alpha \right) =\left( 1+o\left( 1\right) \right) T^{-2\alpha }\log
T+\alpha +o\left( 1\right) ,
\end{equation*
as $T$ tends to infinity; this holds uniformly for $0\leq \alpha \leq
1-\varepsilon .$
\end{theorem}

In fact, this result holds uniformly throughout $0\leq \alpha \leq 1,$ as
later proved by Goldstone.

This result can be used through the identit
\begin{equation*}
\sum_{0<\gamma ,\gamma ^{\prime }\leq T}r\left( \left( \gamma -\gamma
^{\prime }\right) \frac{\log T}{2\pi }\right) w\left( \gamma -\gamma
^{\prime }\right) =\frac{T}{2\pi }\log T\int_{-\infty }^{\infty }F\left(
\alpha \right) \widehat{r}\left( \alpha \right) d\alpha ,
\end{equation*
where
\begin{equation*}
\widehat{r}\left( \alpha \right) =\int_{-\infty }^{\infty }r\left( u\right)
e^{-2\pi i\alpha u}du.
\end{equation*}

Montgomery conjectures that
\begin{equation*}
F\left( \alpha \right) =1+o\left( 1\right)
\end{equation*
for $\alpha \geq 1,$ uniformly in bounded intervals. It can also be
formulated as follows.

\begin{conjecture}[Montgomery]
\label{conjecture_Montgomery}For fixed $\alpha <\beta ,$
\begin{equation*}
\sum_{\substack{ 0<\gamma ,\gamma ^{\prime }<T  \\ 2\pi \alpha /\log
T<\left| \gamma -\gamma ^{\prime }\right| <2\pi \beta /\log T}}1\sim \left(
\int_{\alpha }^{\beta }\left[ 1-\left( \frac{\sin \pi u}{\pi u}\right) ^{2
\right] du+\delta \left( \alpha ,\beta \right) \right) \frac{T}{2\pi }\log T,
\end{equation*
as $T$ goes to infinity. Here $\delta \left( \alpha ,\beta \right) =1$ if
0\in \left[ \alpha ,\beta \right] ,$ $\delta \left( \alpha ,\beta \right) =0$
otherwise.
\end{conjecture}

The proof of Montgomery's theorem is based on the following variant of the
explicit formula.

\begin{lemma}[Montgomery]
\label{lemma_montgomery}If $1<\sigma <2$ and $x\geq 1$ then
\begin{eqnarray*}
\left( 2\sigma -1\right) \sum_{\gamma }\frac{x^{i\gamma }}{\left( \sigma
\frac{1}{2}\right) +\left( t-\gamma \right) ^{2}} &=&-x^{-1/2}\left(
\sum_{n\leq x}\Lambda \left( n\right) \left( \frac{x}{n}\right) ^{1-\sigma
+it}+\sum_{n>x}\Lambda \left( n\right) \left( \frac{x}{n}\right) ^{\sigma
+it}\right) \\
&&+x^{1/2-\sigma +it}\left( \log \tau +O_{\sigma }\left( 1\right) \right)
+O_{\sigma }\left( x^{1/2}\tau ^{-1}\right) ,
\end{eqnarray*
where $\tau =\left| t\right| +2.$ The implicit constants depend only on
\sigma .$
\end{lemma}

In a further development Hughes and Rudnick in \cite{hughes_rudnick02} study
the sums $f\left( x_{1}\right) +\ldots f\left( x_{N}\right) $ where $x_{N}$
denote the scaled zeros and calculate their moments. This is possible to do
if $f$ is sufficiently smooth. They proved the following result. Write the
zeros as $\frac{1}{2}+i\gamma _{j}$ where $\gamma _{j}$ are not assumed
real. Let $f$ be a real-valued even function with smooth Fourier transform
that has compact support. Define
\begin{equation*}
\mathcal{N}_{f}\left( t,T\right) :=\sum_{\gamma _{j}}f\left( \frac{\log T}
2\pi }\left( \gamma _{j}-t\right) \right) .
\end{equation*
(If $f$ were the indicator function of an interval $\left[ -1,1\right] $ and
if all $\gamma _{j}$ are real, then this function would count number of
zeros in the interval $\left[ t-2\pi /\log T,t+2\pi /\log T\right] .$
However, the Fourier transform of this indicator function does not have
compact support.)

Next, choose a weight function $w\left( x\right) ,$ such that $w\geq 0,$
\int w\left( x\right) dx=1,$ and $\widehat{w}\left( x\right) $ is compactly
supported, and define an averaging operator
\begin{equation*}
\left\langle W\right\rangle _{T,H}:=\int_{\mathbb{R}}W\left( t\right)
w\left( \frac{t-T}{H}\right) \frac{dt}{H}.
\end{equation*}

\begin{theorem}[Hughes-Rudnick]
Assume that the averaging window $H=T^{a}$ for $0<a\leq 1,$ and that the
support of $\widehat{f}\left( u\right) =\int f\left( x\right) e^{-2\pi
ixu}dx $ is in $\left( -2a/m,2a/m\right) $ with integer $m\geq 1.$ Then, as
T\rightarrow \infty ,$ the first $m$ moments of $\mathcal{N}_{f},$
\left\langle N_{f}^{m}\right\rangle _{T,H}$ converge to those of a Gaussian
random variable with expectation $\int f\left( x\right) dx$ and variance
\begin{equation*}
\sigma _{f}^{2}=\int \min \left( \left| u\right| ,1\right) \widehat{f}\left(
u\right) ^{2}du.
\end{equation*}
\end{theorem}

They note that a similar result holds in the random matrix theory for
eigenvalues of a unitary random matrix, and that the higher moments converge
to non-Gaussian values. This is again in agreement with the situation in the
random matrix theory.

We will describe the ideas of their proof below in the case when they are
applied to zeros of Dirichlet's L-functions. The key idea is to use a
variant of the explicit formula (\ref{formula_delsarte}).

A generalization of Montgomery results about pair correlations of zeros is
due to Rudnick and Sarnak in \cite{rudnick_sarnak96} who studied statistical
properties of $k$-tuples of zeros. They proved these results for a quite
large class of zeta functions, and we will postpone the discussion of their
results to a later section.

Rudnick and Sarnak proved their results under some restrictive conditions,
similar to conditions imposed by Montgomery. Roughly speaking these
conditions require that the test functions used for evaluation of
correlation functions are very smooth. It is an outstanding problem to prove
that all results about correlations of zeros hold without these restrictive
hypotheses. This is supported by numerical evidence and, if this conjecture
true, we can conclude that the behavior of the zeta zeros is very similar to
the behavior of eigenvalues of random unitary matrices.

\subsection{Dirichlet's L-functions}

Let $\chi \left( n\right) $ denote a multiplicative \emph{character} modulo
a positive integer $q.$ That is, the function $\chi $ maps integers to the
unit circle; it is multiplicative, $\chi \left( nm\right) =\chi \left(
n\right) \chi \left( m\right) ,$ and $\chi \left( n\right) =0$ if $n$ and $q$
are not relatively prime. The character which sends every integer relatively
prime to $q$ to $1$ is called the \emph{principal character} modulo $q$. The
\emph{conductor} of the character is the minimal integer $N$ such that the
character is periodic modulo $N.$ For simplicity, let $q$ be a prime in the
following. In this case the conductor equals $q.$ A character is \emph{odd}
if $\chi \left( -1\right) =-1,$ and \emph{even} if $\chi \left( -1\right)
=1. $

Dirichlet's $L$-function corresponding to the character $\chi $ is defined
by the series
\begin{equation*}
L\left( s,\chi \right) =\sum_{n=1}^{\infty }\frac{\chi \left( n\right) }
n^{s}}.
\end{equation*
This series can be continued to a function which is meromorphic in the whole
complex plane and satisfies a functional equation. Namely, let $\mu =0$ if
\chi $ is even and $\mu =1$ if $\chi $ is odd. Define
\begin{equation*}
\Phi \left( s,\chi \right) =q^{\frac{1}{2}\left( s+\mu \right) }\pi ^{-\frac
1}{2}\left( s+\mu \right) }\Gamma \left( \frac{1}{2}\left( s+\mu \right)
\right) L\left( s,\chi \right) ,
\end{equation*
then the functional equation has the for
\begin{equation*}
\Phi \left( 1-s,\overline{\chi }\right) =\frac{i^{\mu }\sqrt{q}}{\tau \left(
\chi \right) }\Phi \left( s,\chi \right) ,
\end{equation*
where $\tau \left( \chi \right) $ is the Gauss sum:
\begin{equation*}
\tau \left( \chi \right) =\sum_{m=1}^{q}\chi \left( m\right) e^{2\pi im/q}.
\end{equation*
Note that $\left\vert \tau \left( \chi \right) \right\vert =\sqrt{q}.$ For a
proof see Chapter 9 in Davenport \cite{davenport67}.

Finally, note that if $\chi $ is not principal, then $L\left( s,\chi \right) $ is entire.

\subsubsection{Global scale}

For $T>0,$ let $N\left( T,\chi \right) $ denote the number of zeros of
L\left( s,\chi \right) $ with $0<\sigma <1$ and $0\leq t\leq T,$ possible
zeros with $t=0$ or $t=T$ counting one half only. Let

\begin{equation*}
S\left( t,\chi \right) =\frac{1}{\pi }\mathrm{Im}\log L(\frac{1}{2}+it,\chi
).
\end{equation*
Then it can be shown that
\begin{equation*}
\mathcal{N}(T,\chi )=\frac{T}{2\pi }\log \frac{Tq}{2\pi e}-\frac{\chi \left(
-1\right) }{8}+S\left( T,\chi \right) -S\left( 0,\chi \right) +O\left( \frac
1}{1+T}\right) .
\end{equation*
See formula (1.3) in Selberg's paper \cite{selberg46} and for example,
Chapter 16 in Davenport \cite{davenport67}.

If the character is fixed and the imaginary part of the argument is large,
then \ the results for Dirichlet's $L$-functions are similar to those for
Riemann's zeta function.

A different situation arises when the interval $\left[ 0,T\right] $ is fixed
and the character $\chi $ is random. \ This situation was studied by
Selberg, who proved the following results. (See \cite{selberg46}.)

First, for $\left\vert t\right\vert \leq q^{1/4-\varepsilon }$, the
differenc
\begin{equation*}
S\left( t,\chi \right) -\frac{1}{\pi }\mathrm{Im}\sum_{p<q}\frac{\chi \left(
p\right) }{p^{\frac{1}{2}+it}}
\end{equation*
is uniformly bounded in the mean as $q\rightarrow \infty ,$ and $\left\vert
S\left( t,\chi \right) \right\vert $ is of the order of magnitude $\sqrt
\log \log q}$ in the mean. Selberg also established the following theorem.

\begin{theorem}[Selberg]
\label{theorem_selberg_L}For $\left\vert t\right\vert \leq
q^{1/4-\varepsilon },$ we have
\begin{equation*}
\frac{1}{q-2}\sum_{\chi }\left\vert S\left( t,\chi \right) \right\vert ^{2r}
\frac{\left( 2r\right) !}{r!\left( 2\pi \right) ^{2r}}\left( \log \log
q\right) ^{r}+O(\left( \log \log q\right) ^{r-1}),
\end{equation*
where the summation if over all non-principal characters over the base $q.$
\end{theorem}

In other words the distribution of $S\left( t,\chi \right) $ approaches the
distribution of a Gaussian random variable with variance $\frac{1}{2\pi ^{2}
\log \log q.$ In these results $t$ is assumed to be fixed.

\subsubsection{Local Scale}

Hughes and Rudnick in \cite{hughes_rudnick03} study the linear statistics of
low-lying zeros of $L$-functions on the local scale. Hughes and Rudnick
order the zeros $\rho _{i,\chi }=\frac{1}{2}+i\gamma _{i,\chi }$ as follows:
\begin{equation*}
\ldots \leq \mathrm{Re}\gamma _{-2,\chi }\leq \mathrm{Re}\gamma _{-1,\chi
}<0\leq \mathrm{Re}\gamma _{1,\chi }\leq \mathrm{Re}\gamma _{2,\chi }\leq
\ldots
\end{equation*
and define
\begin{equation*}
x_{i,\chi }=\frac{\log q}{2\pi }\gamma _{i,\chi }.
\end{equation*
Then they define
\begin{equation*}
W_{f}\left( \chi \right) =\sum_{i=-\infty }^{\infty }f\left( x_{i,\chi
}\right)
\end{equation*
where $f$ is a rapidly decaying test function.

The question is to understand the behavior of the averages
\begin{equation*}
\left\langle W_{f}^{m}\right\rangle =\frac{1}{q-2}\sum_{\chi \neq \chi
_{0}}W_{f}\left( \chi \right) .
\end{equation*}

The basis for their analysis is a variant of the explicit formula (\re
{formula_delsarte}) that relates a sum over zeros of $L\left( s,\chi \right)
$ to a sum over prime powers. This formula is a particular version of the
formula from Rudnick and Sarnak \cite{rudnick_sarnak96}, which is valid for
zeta functions of all cuspidal automorphic representations (which we will
not define in this paper). Let
\begin{equation*}
\mu \left( \chi \right) =\left\{
\begin{array}{cc}
0, & \text{if }\chi \text{ is even,} \\
1, & \text{if }\chi \text{ is odd,
\end{array
\right.
\end{equation*
and let $h\left( r\right) $ be any even analytic function in the strip
-c\leq \mathrm{Im}r\leq 1+c$ (for $c>0$) such that $\left\vert h\left(
r\right) \right\vert \leq A\left( 1+\left\vert r\right\vert \right)
^{-\left( 1+\delta \right) }$ (for $r\in \mathbb{R},$ $A>0,$ $\delta >0$).
Set $\widehat{h}\left( u\right) =\frac{1}{2\pi }\int h(r)e^{-iru}dr,$ so
that $h\left( r\right) =\int \widehat{h}(u)e^{iru}du$. The
\begin{eqnarray*}
\sum_{j}h\left( \gamma _{j,x}\right) &=&\frac{1}{2\pi }\int_{-\infty
}^{\infty }h\left( r\right) \left( \log q+G_{\chi }\left( r\right) \right) dr
\\
&&-\sum_{n}\frac{\Lambda \left( n\right) }{\sqrt{n}}\widehat{h}\left( \log
n\right) \left( \chi \left( n\right) +\overline{\chi }\left( n\right)
\right) ,
\end{eqnarray*
where
\begin{equation*}
G_{\chi }\left( r\right) =\frac{\Gamma ^{\prime }}{\Gamma }\left( \frac{1}{2
+\mu \left( \chi \right) +ir\right) +\frac{\Gamma ^{\prime }}{\Gamma }\left(
\frac{1}{2}+\mu \left( \chi \right) -ir\right) -\frac{1}{2}\log \pi .
\end{equation*}

By using this formula, Hughes and Rudnick separate $W_{f}\left( \chi \right)
$ into two components. Take $h\left( r\right) =f\left( \frac{\log q}{2\pi
r\right) $, so that $\widehat{h}\left( u\right) =\frac{1}{\log q}\widehat{f
\left( \frac{u}{\log q}\right) .$ We have to impose some conditions on $f$
so that the conditions on $h\left( r\right) $ are satisfied. We say that $f$
is \emph{admissible}, if it is a real, even function whose Fourier transform
$\widehat{f}\left( u\right) :=\int f(r)e^{-2\pi iru}dr$ is compactly
supported, and such that $\left\vert f\left( r\right) \right\vert \leq
A\left( 1+\left\vert r\right\vert \right) ^{-\left( 1+\delta \right) }.$
Then, we get the following decomposition:

\begin{equation*}
W_{f}\left( \chi \right) =\overline{W_{f}}\left( \chi \right)
+W_{f}^{osc}\left( \chi \right) ,
\end{equation*
where
\begin{equation*}
\overline{W_{f}}\left( \chi \right) :=\int_{-\infty }^{\infty }f\left( \frac
\log q}{2\pi }r\right) \left( \log q+G_{\chi }\left( r\right) \right) dr,
\end{equation*
an
\begin{equation*}
W_{f}^{osc}\left( \chi \right) :=-\frac{1}{\log q}\sum_{n}\frac{\Lambda
\left( n\right) }{\sqrt{n}}\widehat{f}\left( \frac{\log n}{\log q}\right)
\left( \chi \left( n\right) +\overline{\chi }\left( n\right) \right) .
\end{equation*}

For large $q,$
\begin{equation*}
\overline{W_{f}}\left( \chi \right) :=\int_{-\infty }^{\infty }f\left(
x\right) dx+O\left( \frac{1}{\log q}\right) ,
\end{equation*
which is asymptotically independent of $\chi .$

First, Hughes and Rudnick show that if $\mathrm{Supp}\left( \widehat{f
\right) \subseteq \left[ -2,2\right] ,$ then
\begin{equation*}
\left\langle W_{f}^{osc}\left( \chi \right) \right\rangle _{q}=O\left( \frac
1}{\log q}\right) .
\end{equation*
Next, they prove the following result.

\begin{theorem}[Hughes and Rudnick]
Let $f$ be an admissible function and assume that
\begin{equation*}
\mathrm{Supp}\left( \widehat{f}\right) \subseteq \left[ -\alpha ,\alpha
\right] ,\text{ }\alpha >0.
\end{equation*
If $m<2/\alpha ,$ then the $m$-th moment of $W_{f}^{osc}$ is \
\begin{equation*}
\lim_{q\rightarrow \infty }\left\langle \left( W_{f}^{osc}\right)
^{m}\right\rangle _{q}=\left\{
\begin{array}{cc}
\frac{m!}{2^{m/2}\left( m/2\right) !}\sigma \left( f\right) ^{m}, & \text{if
}m\text{ is even,} \\
0, & \text{if }m\text{ is odd,
\end{array
\right.
\end{equation*
where
\begin{equation*}
\sigma \left( f\right) ^{2}=\int_{-1}^{1}\left\vert u\right\vert \widehat{f
\left( u\right) ^{2}du.
\end{equation*}
\end{theorem}

They also made a conjecture about the limit when the support of $\widehat{f}$
is not bounded to $\left[ -2/m,2/m\right] $ by performing an analogous
calculation for unitary random matrices (where all moments of the eigenvalue
distribution are known from the work of Diaconis and Shahshahani \cit
{diaconis_shahshahani94}). A consequence of this conjecture is that the
limit of the linear statistic $W_{f}\left( \chi \right) $ is not Gaussian
for high moments.

Hughes and Rudnick used their results to derive some results for the
smallest zero of $L\left( s,\chi \right) .$ In particular they show that for
infinitely many $q$ there are characters $\chi $ such that the imaginary
part of the zero is smaller than $1/4,$ and that for all sufficiently large
q,$ a positive proportion of $L\left( s,\chi \right) $ has a zero smaller
than a certain constant.

\subsection{L-functions for modular forms (The Hecke L-functions)}

A good introductory source for the material in this section is Chapter V of
Milne's book \cite{milne06} about elliptic curves.

\subsubsection{L-functions from modular forms}

\label{section_L_modular_forms}

Let $\Gamma $ is a subgroup of finite index in $SL_{2}\left( \mathbb{Z
\right) $ and let $\mathbb{H}$ denote the upper half-plane $\left\{ z
\mathrm{Im}z>0\right\} .$ The set $\mathbb{H}^{\ast }=\mathbb{H\cup }\left\{
\infty \right\} \cup \mathbb{Q}$ can be made into a Hausdorff topological
space and one can define a continuous action of $SL_{2}\left( \mathbb{Z
\right) $ on $\mathbb{H}^{\ast }$ as an extension of the action of
SL_{2}\left( \mathbb{Z}\right) $ on $\mathbb{H}:
\begin{equation*}
\left(
\begin{array}{cc}
a & b \\
c &
\end{array
\right) f\left( z\right) =f\left( \frac{az+b}{cz+d}\right) .
\end{equation*
The \emph{cusps} are points in $\mathbb{H}^{\ast }\backslash \mathbb{H}$.
One can also show that $\Gamma \backslash \mathbb{H}^{\ast }$ is a compact
Hausdorff space (that is, a compact space in which every two points have
disjoint open neighbourhoods), which is a Riemann surface (that is, it
admits a complex structure). Recall the following definition.

\begin{definition}
A \emph{modular form} for $\Gamma $ of weight $2k$ is a function $\mathbb{H
\rightarrow \mathbb{C}$ such that
\end{definition}

(a) $f$ is holomorphic on $\mathbb{H}$; \

(b) for any $\gamma =\left(
\begin{array}{cc}
a & b \\
c &
\end{array
\right) \in \Gamma ,$ $f\left( \gamma z\right) =\left( cz+d\right)
^{2k}f\left( z\right) ;$

(c) $f$ is holomorphic at the cusps.

In particular, analyticity at infinity implies that a modular form can be
written as
\begin{equation*}
f\left( z\right) =\sum_{n\geq 0}c\left( n\right) q^{n},\text{ where
q=e^{2\pi iz/h},
\end{equation*
where $h$ is the minimal integer $h>0,$ such that $\left(
\begin{array}{cc}
1 & h \\
0 &
\end{array
\right) \in \Gamma .$

A \emph{cusp form} is a modular form which is zero at all the cusps. It can
be written
\begin{equation*}
f\left( z\right) =\sum_{n\geq 1}c\left( n\right) q^{n}.
\end{equation*}

It is usual to write $\mathcal{M}_{2k}\left( \Gamma \right) $ and $\mathcal{
}_{2k}\left( \Gamma \right) $ to denote the vector spaces of all modular and
cusp forms.

Recall that $\Gamma _{0}\left( N\right) $ is a subgroup of $SL_{2}\left(
\mathbb{Z}\right) $ defined as follows
\begin{equation*}
\Gamma _{0}\left( N\right) =\left\{ \left(
\begin{array}{cc}
a & b \\
c &
\end{array
\right) |\text{ }c\equiv 0\left( \mathrm{mod}N\right) \right\} .
\end{equation*}

\begin{definition}
Let $f$ be a cusp form of weight $2k$ for $\Gamma _{0}\left( N\right) ,
\begin{equation*}
f\left( z\right) =\sum_{n\geq 1}c\left( n\right) q^{n}.
\end{equation*
The $L$-series of the cusp form $f$ \ is the Dirichlet series
\begin{equation*}
L\left( f,s\right) =\sum_{n\geq 1}c\left( n\right) n^{-s}.
\end{equation*}
\end{definition}

It is possible to estimate that $\left| c\left( n\right) \right| \leq
Cn^{k}, $ and therefore this series is convergent for $\mathrm{Re}s>k+1.$

Let $\alpha _{N}=\left(
\begin{array}{cc}
0 & -1 \\
N &
\end{array
\right) .$ The
\begin{equation*}
\alpha _{N}\left(
\begin{array}{cc}
a & b \\
c &
\end{array
\right) \alpha _{N}^{-1}=\left(
\begin{array}{cc}
d & -c/N \\
-Nb &
\end{array
\right) ,
\end{equation*
and so the conjugation by $\alpha _{N}$ preserves $\Gamma _{0}\left(
N\right) .$ This implies the following fact about cusp forms. Define
operator $w_{N}$ by the following formula:
\begin{equation*}
\left( w_{N}f\right) \left( z\right) :=\left( \sqrt{N}z\right) ^{2k}f\left(
-1/z\right) .
\end{equation*
Then $w_{N}$ preserves $\mathcal{S}_{2k}\left( \Gamma _{0}\left( N\right)
\right) $ and $w_{N}^{2}=1.$ Hence the only eigenvalues of $w_{N}$ are $\pm
1,$ and $\mathcal{S}_{2k}\left( \Gamma _{0}\left( N\right) \right) $ is a
direct sum of the corresponding eigenspaces, $\mathcal{S}_{2k}=\mathcal{S
_{2k}^{+1}+\mathcal{S}_{2k}^{-1}$.

It turns out that the zeta functions corresponding to these eigenvector cusp
forms satisfy functional equations analogous to the functional equation of
the Riemann's zeta.

\begin{theorem}[Hecke]
Let $f\in \mathcal{S}_{2k}\left( \Gamma _{0}\left( N\right) \right) $ be a
cusp form in the $\varepsilon $-eigenspace, $\varepsilon =1$ or $-1.$ Then
f $ extends analytically to a holomorphic function on the whole complex
plane, and satisfies the functional equation
\begin{equation*}
\Lambda \left( s,f\right) =\varepsilon \left( -1\right) ^{k}\Lambda \left(
k-s,f\right) ,
\end{equation*
where
\begin{equation*}
\Lambda \left( s,f\right) =N^{s/2}\left( 2\pi \right) ^{-s}\Gamma \left(
s\right) L\left( f,s\right) .
\end{equation*}
\end{theorem}

Moreover, under certain conditions, which we explain below, these zeta
functions can be written as a Euler product over primes.

Namely, the Hecke operators $T\left( n\right) $ ($n\geq 1$) are linear
operators on $\mathcal{S}_{2k}\left( \Gamma _{0}\left( N\right) \right) .$
We skip their definition here, which can be found in Section V.4 of Milne
\cite{milne06} or Section VII.5 of Serre \cite{serre73} , and only list
their properties.

\begin{theorem}[Hecke]
The maps $T\left( n\right) $ have the following properties: \newline
(a) $T\left( mn\right) =T\left( m\right) T\left( n\right) $ if $\gcd \left(
m,n\right) =1;$\newline
(b) $T\left( p\right) T\left( p^{r}\right) =T\left( p^{r+1}\right)
+p^{2k-1}T\left( p^{r-1}\right) $ if $p$ does not divide $N;$\newline
(c) $T\left( p^{r}\right) =T\left( p\right) ^{r},$ $r\geq 1,$ if $p|N;
\newline
(d) all $T\left( n\right) $ commute.
\end{theorem}

Since the Hecke operators commute we can look for the functions which are
eigenfunctions for all of them. Then, it turns out that these eigenfunctions
have Euler products.

\begin{theorem}[Hecke]
Let $f$ be a cusp form of weight $2k$ for $\Gamma _{0}\left( N\right) $ that
is simultaneously an eigenvector for all $T\left( n\right) ,$ say $T\left(
n\right) f=\lambda \left( n\right) f,$ and let
\begin{equation*}
f=\sum_{n\geq 1}c\left( n\right) q^{n},\text{ }q=e^{2\pi iz}.
\end{equation*
Let $c\left( 1\right) =1.$ Then, (i) coefficients of the series are
eigenvalues of the Hecke operators,
\begin{equation*}
c\left( n\right) =\lambda \left( n\right)
\end{equation*}
and (ii)
\begin{equation*}
L\left( s,f\right) =\prod_{p|N}\frac{1}{1-c\left( p\right) p^{-s}
\prod_{p\nmid N}\frac{1}{1-c\left( p\right) p^{-s}+p^{2k-1-s}}.
\end{equation*}
\end{theorem}

In particular, $\mathcal{S}_{12}\left( \Gamma _{0}\left( 1\right) \right) $
has dimension $1,$ and therefore it is generated by a single function, which
is called the $\Delta $-function:
\begin{equation*}
\Delta \left( q\right) =q\prod_{1}^{\infty }\left( 1-q^{n}\right) ^{24}=\sum
\tau \left( n\right) q^{n},
\end{equation*
where $\tau \left( n\right) $ is called the Ramanujan $\tau $-function. It
follows that the $L$-function associated to the $\Delta $-function has both
a functional equation and the Euler product property.

In a more general situation, if we wish to find forms that have both a
functional equation and a Euler product, then we must overcome the obstacle
that operators $w_{N}$ and $T\left( n\right) $ do not always commute.
However, this obstacle can be circumvented and it can be proved that such
modular forms do exist. They are called \emph{newforms. }

In summary, the L-functions of modular newforms have both a functional
equation and the Euler product property.

\subsubsection{L-functions from Maass forms}

A nice source for the material in this section is the lecture notes by Liu
\cite{liu07}. A lot of additional information about Maass forms can be found
in the book by Iwaniec \cite{iwaniec02}.

Modular forms are holomorphic and so are comparably rare animals. It is not
immediately clear how to compute them. We can try to relax the assumption of
complex analyticity, and in this way we come to the concept of a Maass form.

\begin{definition}
A smooth function $f\neq 0$ is called a \emph{Maass form} for group $\Gamma
, $ if \newline
(i) for all $g\in \Gamma $ and all $z\in \mathbb{H},$ $f\left( gz\right)
=f\left( z\right) ;$\newline
(ii) $f$ is an eigenfunction of the non-Euclidean Laplace operator:
\begin{equation*}
-y^{2}\left( \frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}
\partial y^{2}}\right) f=\lambda f,
\end{equation*
and\newline
(iii) there exists a positive integer $N,$ such that
\begin{equation*}
f\left( z\right) \ll y^{N},\text{ }y\rightarrow \infty .
\end{equation*}
\end{definition}

Note that it is relatively easy to generate Maass forms as eigenfunctions of
the Laplace operator on a fundamental domain of the group $\Gamma .$

\begin{definition}
A Maass form f is said to be a \emph{cusp form} if the equality
\begin{equation*}
\int_{0}^{1}f\left( z+b\right) db=0
\end{equation*
holds for all $z\in \mathbb{H}$.
\end{definition}

A Maass form $f$ is call \emph{odd} if $f\left( -x+iy\right) =-f\left(
x+iy\right) ,$ and \emph{even} if $f\left( -x+iy\right) =f\left( x+iy\right)
.$

One example of a Maass form is provided by the Eisenstein series:
\begin{eqnarray}
E\left( z,s\right) &=&\frac{\Gamma \left( s\right) }{\pi ^{s}}\sum_{\left(
m,n\right) \neq \left( 0,0\right) }\frac{y^{s}}{\left\vert mz+n\right\vert
^{2s}}  \notag \\
&=&\pi ^{-s}\Gamma \left( s\right) \zeta \left( 2s\right) \left( 2y^{s}+\sum
_{\substack{ \left( c,d\right) =1  \\ c\neq 0}}\frac{y^{s}}{\left\vert
cz+d\right\vert ^{2s}}\right)  \notag \\
&=&\pi ^{-s}\Gamma \left( s\right) \zeta \left( 2s\right) \sum_{\gamma \in
\Gamma _{\infty }\backslash \Gamma }\left( \mathrm{Im}\left( \gamma z\right)
\right) ^{s},  \notag \\
\text{where }\Gamma _{\infty } &=&\left\{ \gamma \in \Gamma |\gamma \infty
=\infty \right\} .
\end{eqnarray}

(The factor $\pi ^{-s}\Gamma \left( s\right) $ is introduced to get a
simpler functional equation.) It can be checked that this series is an
eigenfunction of the Laplace operator with eigenvalue $s\left( 1-s\right) ,$
and it turns out that it possess a nice functional equation. Eisenstein
series can also be defined for different subgroups $\Gamma $ and different
cusps.

From the spectral point of view, Eisenstein series correspond to the
continuous part of the spectrum of the Laplacian. There are also other forms
that correspond to the discrete part of the spectrum.

By expanding a Maass form in Fourier series and taking the Fourier
coefficients as the coefficients of a Dirichlet series, one can construct
new L-functions that satisfy a functional equation. The construction is
similar to the case of modular forms.

\begin{theorem}
Let $f$ be a Maass form with eigenvalue $1/4+\nu ^{2}.$ Let $\varepsilon =0$
or $1$ depending on whether $f$ is even or odd. Let
\begin{equation*}
\Lambda \left( s,f\right) =\pi ^{-s}\Gamma \left( \frac{s+\varepsilon +i\nu
}{2}\right) \Gamma \left( \frac{s+\varepsilon -i\nu }{2}\right) L\left(
s,f\right) .
\end{equation*
Then $\Lambda \left( s,f\right) $ is an entire function that satisfies
\begin{equation*}
\Lambda \left( s,f\right) =\left( -1\right) ^{\varepsilon }\Lambda \left(
1-s,f\right) .
\end{equation*}
\end{theorem}

Since the Laplace operator commutes with Hecke operators, therefore they can
be simultaneously diagonalized.

For example, the Eisenstein series $E\left( z,\frac{1}{2}+it\right) $ is an
eigenfunction of all Hecke operators with an eigenvalue
\begin{equation*}
\eta _{t}\left( n\right) =\sum_{ad=n}\left( \frac{a}{d}\right) ^{it}.
\end{equation*}

The Maass forms which are eigenfunctions of Hecke operators lead to
L-functions with a Euler product property and in many respects the theory is
parallel to the theory of $L$-functions which are obtained from modular
newforms.

In particular, for the L-series corresponding to the Eisenstein series
E_{r}=E\left( z,\frac{1}{2}+ir\right) ,$ one can check that
\begin{equation}
L\left( s,E_{r}\right) =\zeta \left( s+ir\right) \zeta \left( s-ir\right) .
\label{L_for_Eisenstein}
\end{equation}

\subsubsection{Statistical properties of zeros: Global scale}

Let
\begin{equation*}
S\left( t,\psi \right) :=\frac{1}{\pi }\arg L\left( \frac{1}{2}+it,\psi
\right) ,
\end{equation*
where $\psi $ is the Maass form with eigenvalue $\lambda \left( \psi \right)
$ and $L$ is the corresponding $L$-function. $S\left( t,\psi \right) $ is
related to the number of zeros of $L$ in the critical strip in the same way
as the usual $S\left( t\right) $ function is related to the number of zeros
of Riemann's zeta function.

For this function $S\left( t,\psi \right) ,$ Hejhal and Luo \cit
{hejhal_luo97} proved analogues of Selberg's results for the function
S\left( t\right) .$

For the Eisenstein series $E_{r}=E\left( z,\frac{1}{2}+ir\right) ,$ formula
\ref{L_for_Eisenstein}) implies that
\begin{equation*}
S\left( t,E_{r}\right) =S\left( r+t\right) -S\left( r-t\right) ,
\end{equation*
and the results about the statistical properties of $S\left( t,E_{r}\right) $
are reduced to the question about the statistical properties of $S\left(
t\right) .$

For the case of Maass cusp forms with eigenvalues from the discrete part of
the spectrum, Hejhal and Luo proved the following result.

Let $S_{j}\left( t\right) :=S\left( t,\psi _{j}\right) $ where $\psi _{j}$
has an eigenvalue $\lambda _{j}=\frac{1}{4}+r_{j}^{2}.$ Since $\psi
_{j}\left( z\right) $ is periodic along the real axis, it has a Fourier
series, and it is known that it can be written as follows:
\begin{equation*}
\psi _{j}\left( x+iy\right) =\sum_{n\neq 0}\rho _{j}\left( n\right) \sqrt{y
K_{ir_{j}}\left( 2\pi \left\vert n\right\vert y\right) e^{2\pi inx},
\end{equation*
where $K_{\nu }\left( z\right) $ are the standard Bessel functions. Let $\nu
_{j}\left( n\right) =\rho _{j}\left( n\right) /\sqrt{\cosh \pi r_{j}}.$ The
numbers $\nu _{j}\left( 1\right) $ will be used as weights in the limiting
procedure. One knows that they are $O\left( \sqrt{r_{j}}\right) $ and
\begin{equation*}
\frac{1}{T^{2}}\sum_{r_{j}\leq T}\left\vert \nu _{j}\left( 1\right)
\right\vert ^{2}=\frac{1}{\pi ^{2}}+O\left( \frac{\log T}{T}\right) .
\end{equation*
Since by the Weyl law, the number of $r_{j}$ below $T$ is proportionate to
T^{2},$ these weights can be thought as having bounded magnitude and not too
sparse.

Now we formulate one of the results in this paper (Theorem 3 in \cit
{hejhal_luo97}).

\begin{theorem}[Hejhal-Luo]
Let $h>0$ and $t>0$ be fixed. Then we have
\begin{equation*}
\lim_{T\rightarrow \infty }\frac{1}{\left( 2HT\right) }\sum_{\left\vert
r_{j}-T\right\vert \leq H}\frac{\pi ^{2}\left\vert \nu _{j}\left( 1\right)
\right\vert ^{2}}{2}\frac{\left( S_{j}\left( t\right) \right) ^{n}}{\left(
\log \log T\right) ^{n/2}}=C_{n},
\end{equation*
where $C_{n}$ are moments of the Gaussian distribution.
\end{theorem}

The main new ingredient in the proof is the following lemma (Lemma 1 in \cit
{hejhal_luo97}).

\begin{lemma}
Let $T\geq 3,$ $\beta >0,$ $\frac{1}{2}>\varepsilon >0,$ and let $\max
\left( n_{1},n_{2}\right) \leq T^{1-\varepsilon }.$ Then
\begin{equation*}
\sum_{j=1}^{\infty }e^{-\beta \left( T-r_{j}\right) ^{2}}\nu _{j}\left(
n_{1}\right) \overline{\nu _{j}\left( n_{2}\right) }=\frac{2\pi ^{-3/2}}
\sqrt{\beta }}T\delta _{n_{1},n_{2}}+O_{\beta ,\varepsilon }\left( d\left(
n_{1}\right) d\left( n_{2}\right) \left( \log T\right) ^{2}\right) ,
\end{equation*}
\end{lemma}

where $d\left( n\right) $ denotes the number of divisors of $n.$

The proof of this Lemma is based on the so-called Kuznetsov trace formula.

\subsubsection{Local scale}

\label{section_Rudnick_Sarnak}

Rudnick and Sarnak \cite{rudnick_sarnak96} extended the results of
Montgomery to zeta functions that arise from modular forms and from Maass
forms. In fact, they work in greater generality and study the zeta functions
that arise from the general automorphic cuspidal representations of $GL_{m}
. The Hecke modular L-functions correspond to the case $m=2.$ Their main
tool is the following explicit formula, which we formulate for the case of
the Hecke L-functions. Let

\begin{eqnarray*}
L\left( s,f\right) &=&\prod_{p|N}\frac{1}{1-c\left( p\right) p^{-s}
\prod_{p\nmid N}\frac{1}{1-c\left( p\right) p^{-s}+p^{2k-1-s}} \\
&=&\prod L_{p}\left( s,f\right) .
\end{eqnarray*
where
\begin{equation*}
L_{p}\left( s,f\right) =\frac{1}{\left( 1-\alpha _{1}(p)p^{-s}\right) \left(
1-\alpha _{2}(p)p^{-s}\right) },
\end{equation*
with the convention that for $p|N$ one of $\alpha _{i}\left( p\right) $ is
zero. Let $a\left( p^{k}\right) =\alpha _{1}\left( p\right) ^{k}+\alpha
_{2}\left( p\right) ^{k},$ and define $b\left( n\right) =\Lambda \left(
n\right) a\left( n\right) .$ Then
\begin{equation*}
\frac{L^{\prime }}{L}=-\sum_{n=1}^{\infty }\frac{b\left( n\right) }{n^{s}}.
\end{equation*}

\begin{theorem}[Rudnick and Sarnak]
Let $\widehat{h}\in C_{c}^{\infty }\left( \mathbb{R}\right) $ be a smooth
compactly supported function, and let $h\left( r\right) =\int \widehat{h
\left( u\right) e^{iru}du$. Then
\begin{eqnarray}
\sum h\left( \gamma \right) &=&\frac{1}{2\pi }\int_{-\infty }^{\infty
}h\left( r\right) \left( \log Q+\sum_{j}\left[ \frac{\Gamma ^{\prime }}
\Gamma }\left( \frac{1}{2}+\mu _{j}+ir\right) +\frac{\Gamma ^{\prime }}
\Gamma }\left( \frac{1}{2}+\mu _{j}-ir\right) \right] \right) dr  \notag \\
&&-\sum_{n=1}^{\infty }\left( \frac{b\left( n\right) }{\sqrt{n}}\widehat{h
\left( \log n\right) +\frac{\overline{b\left( n\right) }}{\sqrt{n}}\widehat{
}\left( -\log n\right) \right) ,  \label{formula_Rudnick_Sarnak}
\end{eqnarray
where $\mu _{j}$ are some parameters that depend on the form $f,$ and $Q$ is
the conductor of the form.
\end{theorem}

By using this result and estimates on the size of coefficients $b\left(
n\right) $, Rudnick and Sarnak proved a generalization of the Montgomery
theorem. Their result is valid not only for the Riemann zeta function, but
also for Dirichlet $L$-functions, for Hecke modular $L$-functions and for $L
-functions that correspond to automorphic cuspidal representations of
GL_{3}.$ We formulate it for Hecke modular $L$-functions.

Consider the class of smooth test functions $F(x_{1},\ldots ,x_{n})$ that
satisfy the following conditions:

\textbf{TF 1}. $F(x_{1},\ldots ,x_{n})$ is symmetric.

\textbf{TF 2}. $F(x+t\left( 1,\ldots ,1\right) )=F\left( x\right) $ for all
t\in \mathbb{R}$.

\textbf{TF 3}. $F\left( x\right) \rightarrow 0$ rapidly as $\left| x\right|
\rightarrow \infty $ in the hyperplane $\sum x_{j}=0.$

If $B_{N}$ is a set of $N$ numbers $x_{1},\ldots ,x_{N},$ then the $n$-level
correlation sum is defined by
\begin{equation*}
R_{N}\left( B_{N},F\right) =\frac{n!}{N}\sum_{\substack{ S\subset B_{N}  \\
\left| S\right| =n}}F\left( S\right) .
\end{equation*}

Define the $n$-level correlation density by
\begin{equation*}
W_{n}\left( x_{1},\ldots ,x_{n}\right) =\det \left( K\left(
x_{i}-x_{j}\right) \right) ,\text{ }K\left( x\right) =\frac{\sin \pi x}{\pi
}.
\end{equation*}

\begin{theorem}
Assume the Riemann hypothesis for the zeros of $L\left( s,f\right) .$ Let
F\left( x_{1},\ldots ,x_{N}\right) $ satisfy \textbf{TF 1, 2, 3} and in
addition assume that $\widehat{F}\left( \xi \right) $ is supported in
\sum_{j}\left| \xi _{j}\right| <1.$ Then,
\begin{equation*}
R_{n}\left( B_{N},F\right) \rightarrow \int F\left( x\right) W_{n}\left(
x\right) \delta \left( \frac{x_{1}+\ldots +x_{n}}{n}\right) dx_{1}\ldots
dx_{n}
\end{equation*
as $N\rightarrow \infty .$
\end{theorem}

Rudnick and Sarnak mention that the result can probably be proven for
functions $F$ with the Fourier transform supported in $\sum_{j}\left| \xi
_{j}\right| <2$ by an improvement of their method, and conjecture that it
holds without any assumption on the support of $\widehat{F}\left( \xi
\right) .$

\subsection{ Elliptic curve zeta functions}

The main source for this section is the book \cite{milne06} \ by Milne.
Consider an elliptic curve
\begin{equation*}
E:Y^{2}Z=X^{3}+aXZ^{2}+bZ^{3},
\end{equation*
where $a$ and $b$ are integer, and assume that $\left\vert \Delta
\right\vert =\left\vert 4a^{3}+27b^{2}\right\vert $ cannot be made smaller
by a change of variable $X\rightarrow X/c^{2},\,Y\rightarrow Y/c^{3}.$ This
equation is called \emph{minimal}. The equation
\begin{equation*}
\overline{E}:Y^{2}Z=X^{3}+\overline{a}XZ^{2}+\overline{b}Z^{3},
\end{equation*
with $\overline{a}$ and $\overline{b}$ the images of $a$ and $b$ in $\mathbb
F}_{p}$ (the finite field with $p$ elements) is called \emph{the reduction
of }$E$\emph{\ modulo }$p.$ (It is assumed here that $p\neq 2,3.$ In the
case when $p$ is $2$ or $3$, a somewhat different notion of the minimal
equation is needed.) Let $N_{p}$ is the number of solutions of this equation
in $\mathbb{F}_{p}.$

There are four possible cases:

(a) \textbf{Good reduction}. $\overline{E}$ is an elliptic curve. (That is,
the determinant does not vanish and therefore the curve is smooth.) This
happens if $p\neq 2$ and $p$ does not divide $\Delta .$ In this case we
define the number of points of $\overline{E}$ by $N_{p}.$

(b) \textbf{Cuspidal, or additive, reduction.} This is the case in which the
reduced curve has a cusp. For $p\neq 2,3$, this case occurs exactly when
p|4a^{3}+27b^{2}$ and $p|-2ab.$

(c) \textbf{Nodal, or multiplicative, reduction.} The reduced curve has a
node. For $p\neq 2,3$, it occurs exactly when $p|4a^{3}+27b^{2}$, $p\nmid
-2ab.$

\qquad (c1) \textbf{Split case.} The tangents at the node are rational over
\mathbb{F}_{p}.$ This happens when $-2ab$ is a square in $\mathbb{F}_{p}$.

\qquad (c2) \textbf{Non-split case.} The tangents at the node are not
rational over $\mathbb{F}_{p}.$This occurs when $-2ab$ is not a square in
\mathbb{F}_{p}$.

The names additive and multiplicative refer to the group of points on the
reduced curve, which in these cases is isomorphic either to $\left( \mathbb{
}_{p},+\right) ,$ or $\left( \mathbb{F}_{p}^{\ast },\times \right) .$

We define the $L$ function associated with the elliptic curve $E$ as follows
\begin{equation*}
L\left( s,E\right) :=\prod_{p}\frac{1}{L_{p}\left( p^{-s}\right) }.
\end{equation*}

Here, the local factors $L_{p}\left( T\right) $ are defined as follows:
\begin{equation*}
L_{p}\left( T\right) =\left\{
\begin{array}{cc}
1-a_{p}T+pT^{2}, & \text{if }p\text{ is good, with }a_{p}=p+1-N_{p}, \\
1-T & \text{if }E\text{ has split multiplicative reduction,} \\
1+T & \text{if }E\text{ has non-split multiplicative reduction,} \\
1 & \text{if }E\text{ has additive reduction.
\end{array
\right.
\end{equation*}

Let $S$ be the (finite) set of primes with bad reduction. Then we can also
write
\begin{eqnarray*}
L\left( s,E\right) &:&=\prod_{p\in S}\frac{1}{L_{p}\left( p^{-s}\right)
\prod_{p\notin S}\frac{1}{1+\left( N_{p}-p-1\right) p^{-s}+p^{1-2s}} \\
&=&\prod_{p\in S}\frac{1}{L_{p}\left( p^{-s}\right) }\prod_{p\notin S}\frac{
}{\left( 1-\alpha _{p}p^{-s}\right) \left( 1-\beta _{p}p^{-s}\right) }.
\end{eqnarray*}

The Hasse-Weil conjecture says that $L\left( s,E\right) $ can be
analytically continued to a meromorphic function on the whole of $\mathbb{C}$
and satisfies a functional equation. The recent work by Wiles and others
confirmed this conjecture by showing that all elliptic curves are
\textquotedblleft modular\textquotedblright , in particular, their $L
-functions arise from modular forms. To a certain extent, this result
reduces the study of the elliptic $L$-functions to the study of the Hecke
modular $L$-functions.

It is known that the numbers $a_{p}$ do not exceed $2\sqrt{p}$ in absolute
value. For a fixed elliptic curve and different primes $p,$ these numbers
are believed to be distributed on the interval $\left[ -2\sqrt{p},2\sqrt{p
\right] $ according to the semicircle distribution but this is not proven.
In fact, this conjecture is related to the Birch-Swinnerton conjecture that
states that
\begin{equation*}
L\left( s,E\right) \sim C\left( s-1\right) ^{r}\text{ as }s\rightarrow 1,
\end{equation*
where $r$ is the rank of the group of rational points on $E,$ and $C$ is a
certain predicted constant.

A stronger statement, which can still be true, is that
\begin{equation*}
\prod_{p\notin S}\frac{N_{p}}{p}=\prod_{\substack{ p\notin S  \\ p\leq x}
\left( 1-\frac{a_{p}-1}{p}\right) =C\left( \log x\right) ^{r}.
\end{equation*}

It is possible to construct zeta functions for other nonsingular projective
varieties and the conjecture by Hasse and Weil states that these zeta
functions satisfy a functional equation (and the Riemann hypothesis).
However, apparently not much is known beyond the cases of projective spaces
and elliptic curves.

\section{Selberg's zeta functions for compact and non-compact manifolds}

The main source for this section is Hejhal's book \cite{hejhal76}.

\subsection{Selberg's zeta function and trace formula}

Let $M$ be a compact Riemann surface of genus $g\geq 2.$ Then, $M$ can be
identified with a quotient space $\Gamma \backslash \mathbb{H},$ where
\mathbb{H}$ is the upper half-plane and $\Gamma $ is a discrete subgroup of
SL_{2}\left( R\right) $. We assume that $\mathbb{H}$ has the Poincare metric
$\left\vert dz\right\vert /y$ with the area element $dxdy/y^{2},$ and
therefore it has the constant negative curvature. This metric is naturally
projected on the surface $M.$

This is not the most general situation of interest since most of the
quotient spaces $\Gamma \backslash \mathbb{H}$ occurring in arithmetic
applications have cusps and therefore are non-compact. However, the theory
is most clear and transparent for the compact surfaces.

The Laplace operator on $M$ can be defined by the following formula.
\begin{equation*}
-\Delta :u\rightarrow -y^{2}\left( u_{xx}+u_{yy}\right) .
\end{equation*
It can be shown that this operator has a discrete set of non-positive
eigenvalues
\begin{equation*}
0=\lambda _{0}<\lambda _{1}\leq \lambda _{2}\leq \ldots ,
\end{equation*
and the only point of accumulation of these eigenvalues is $\infty .$

Let us define
\begin{equation*}
r_{n}=\left\{
\begin{array}{cc}
\sqrt{\lambda _{n}-\frac{1}{4}}, & \text{if }\lambda _{n}\geq \frac{1}{4},
\\
i\sqrt{-\lambda _{n}+\frac{1}{4}}, & \text{if }\lambda _{n}<1/4
\end{array
\right.
\end{equation*
so that $\lambda _{n}=\frac{1}{4}+r_{n}^{2}.$

Also let $\overline{m}=\max \left\{ k:\lambda _{k}<1/4\right\} .$

Let $\mathcal{G}\left( M\right) $ be the set of all closed geodesics on $M$,
and let $\mathcal{P}\left( M\right) $ be the subset of all prime closed
geodesics (that is, the closed geodesics that cannot be represented as a
non-trivial multiple of another closed geodesic). It is known that it is a
countable set, which we can order by the lengths of its elements. Closed
geodesics correspond to hyperbolic elements of the group $\Gamma $ (that is,
the elements of $\Gamma $ with the trace outside of $[-2,2]$) up to
conjugacy of these elements. If $P\in \Gamma $ corresponds to a geodesic
\gamma ,$ then $\gamma $ is prime if and only if there is no $P_{0}\in
\Gamma $ such that $P=P_{0}^{k}$ for an integer $k>1.$

If $l\left( \gamma \right) $ denotes the length of the geodesic $\gamma ,$
then we set
\begin{equation*}
\left\vert \gamma \right\vert :=e^{l\left( \gamma \right) }.
\end{equation*
If $P$ is the hyperbolic element of $\Gamma $ that corresponds to $\gamma ,$
then
\begin{equation*}
\left\vert \gamma \right\vert ^{1/2}+\left\vert \gamma \right\vert
^{-1/2}=\left\vert \mathrm{Tr}P\right\vert .
\end{equation*
We will also write $\left\vert \gamma \right\vert =N\left[ P\right] $
(meaning norm of $P$).

The Selberg trace formula relates sums over eigenvalues $\lambda _{k}$ to
sums over hyperbolic elements (geodesics) $\left[ P\right] $. Let $h\left(
u\right) $ be a function which (i) is analytic in the strip $\left\vert
\mathrm{Im}u\right\vert \leq 1/2+\delta ,$ (ii) is even: $h(u)=h\left(
-u\right) ,$ and (iii) declines sufficiently fast in the strip: $\left\vert
h\left( u\right) \right\vert =O\left( \left( 1+\left\vert \mathrm{Re
u\right\vert \right) ^{-2-\delta }\right) $. This ensures absolute
convergence in the following formulas.

Let $\widehat{h}\left( t\right) =\frac{1}{2\pi }\int h\left( u\right)
e^{-itu}du.$

Then
\begin{eqnarray}
\sum_{n=0}^{\infty }h\left( r_{n}\right) &=&\frac{\mu \left( F\right) }{2\pi
}\int_{\mathbb{R}}rh\left( r\right) \tanh \left( \pi r\right) dr
\label{formula_Selberg_trace} \\
&&+\sum_{\left[ T\right] }\frac{\ln N\left[ T_{0}\right] }{N\left[ T\right]
^{1/2}-N\left[ T\right] ^{-1/2}}\widehat{h}\left( \ln N\left[ T\right]
\right) ,  \notag
\end{eqnarray
where the sum is over all distinct conjugacy classes of hyperbolic elements
\left[ T\right] $ and $\left[ T_{0}\right] $ is the primitive element for
T, $ $T=T_{0}^{k}.$

It is instructive to compare this formula with formula (\re
{formula_Rudnick_Sarnak}). Since it resembles the explicit formulas from
number theory, it is natural to seek for a corresponding zeta functions.

We define the \emph{Selberg's zeta function} as follows:
\begin{equation}
Z\left( s\right) =\prod_{\gamma \in \mathcal{P}\left( M\right)
}\prod_{k=0}^{\infty }\left( 1-\left\vert \gamma \right\vert ^{-s-k}\right)
\text{ }\mathrm{Re}s>1.  \label{zeta_Selberg}
\end{equation}

It turns out that the Selberg's zeta function is closely related to the
eigenvalues of the Laplace operator on $M.$

\begin{theorem}[Hejhal-Selberg]
(a) $Z\left( s\right) $ is actually an entire function;\newline
(b) Let $\beta $ be a real number $\geq 2.$ For all $s,$ the following
identity holds:
\begin{eqnarray*}
\frac{1}{2s-1}\frac{Z^{\prime }\left( s\right) }{Z\left( s\right) } &=&\frac
1}{2\beta }\frac{Z^{\prime }\left( \frac{1}{2}+\beta \right) }{Z\left( \frac
1}{2}+\beta \right) }+\sum_{n=0}^{\infty }\left[ \frac{1}{r_{n}^{2}+\left( s
\frac{1}{2}\right) ^{2}}-\frac{1}{r_{n}^{2}+\beta ^{2}}\right] \\
&&+\frac{\mu \left( F\right) }{2\pi }\sum_{k=0}^{\infty }\left[ \frac{1}
\beta +\frac{1}{2}+k}-\frac{1}{s+k}\right] ,
\end{eqnarray*
where $\mu \left( F\right) $ is the area of the fundamental region of the
group $\Gamma .$\newline
(c) $Z\left( s\right) $ has ``trivial'' zeros $s=-k,$ $k\geq 1,$ with
multiplicity $\left( 2g-2\right) \left( 2k+1\right) ;$\newline
(d) $s=0$ is a zero of multiplicity $2g-1;$\newline
(e) $s=1$ is a zero of multiplicity $1;$\newline
(f) the nontrivial zeros of $Z\left( s\right) $ are located at $\frac{1}{2
\pm ir_{n}.$
\end{theorem}

Since all but a finite number of eigenvalues are greater than $1/4$ hence
all but a finite number of $r_{n}$ is real and therefore the claim (f)
implies that all but a finite number of zeros of $Z\left( s\right) $ are
located on the line $\mathrm{Im}z=1/2.$

The formula in claim (b) of this theorem follows from Selberg's trace
formula and it can be thought as a functional equation for the logarithmic
derivative of $Z\left( s\right) $. In particular, it implies the functional
equation for the zeta functions itself.

\begin{theorem}[Hejhal-Selberg]
Selberg's zeta function satisfies the following functional equation:
\end{theorem}

\begin{equation*}
Z\left( s\right) =Z\left( 1-s\right) \exp \left[ \mu \left( F\right)
\int_{0}^{s-\frac{1}{2}}v\tan \left( \pi v\right) dv\right] .
\end{equation*}

\subsection{Statistics of zeros}

The number of zeta zeros in a long interval has the following asymptotics:
\begin{equation*}
N\left[ k:0\leq r_{k}\leq T\right] =\frac{\mu \left( F\right) }{4\pi
T^{2}+S\left( T\right) +E\left( T\right) ,
\end{equation*
where
\begin{equation*}
S\left( T\right) =\frac{1}{\pi }\arg Z\left( \frac{1}{2}+iT\right) ,
\end{equation*
and
\begin{equation*}
E\left( T\right) =O\left( 1\right) =2c\int_{0}^{T}t\left[ \tanh \left( \pi
t\right) -1\right] dt-\left( \overline{m}+1\right) .
\end{equation*}

In other words, the number of zeros in a unit interval is $\sim cT.$ In
comparison, for Riemann's zeta we have $\sim c\log T$ zeros in the unit
interval.

It is known that
\begin{equation*}
S\left( T\right) =O\left[ \frac{T}{\log T}\right] ,\text{ and }S\left(
T\right) =\Omega _{\pm }\left[ \left( \frac{\log T}{\log \log T}\right)
^{1/2}\right] .
\end{equation*
(Recall that the notation $f\left( x\right) =\Omega _{+}\left[ g(x)\right] $
means that $\lim \sup \left( \frac{f\left( x\right) }{g\left( x\right)
\right) >0,$ and $f\left( x\right) =\Omega _{-}\left[ g(x)\right] $ means
that $\lim \inf \left( \frac{f\left( x\right) }{g\left( x\right) }\right)
<0. $)

It was found that it is difficult to generalize the results concerning the
moments of the $S\left( x\right) $ function to the Selberg's zeta. Since
these results are essential for the study of statistical properties of zeta
zeros, there is a stumbling block here.

Selberg managed to generalize some of these statistical results for a
particular choice of the group $\Gamma .$

Let $p\geq 3$ be a prime and $A$ be a quadratic non-residue modulo $p.$
Define
\begin{equation*}
\Gamma =\Gamma \left( A,p\right) =\left\{ \left(
\begin{array}{cc}
y_{0}+y_{1}\sqrt{A} & y_{2}\sqrt{p}+y_{3}\sqrt{Ap} \\
y_{2}\sqrt{p}-y_{3}\sqrt{Ap} & y_{0}-y_{1}\sqrt{A
\end{array
\right) ;\text{ }y_{0},y_{1},y_{2},y_{3}\text{ are integer.}\right\}
\end{equation*
and call it a \emph{quaternion group.}

It turns out that $\Gamma \left( A,p\right) $ has no parabolic elements,
(that is, it has no group elements with trace equal to $\pm 2,$ except $\pm
I $). Moreover, it corresponds to a compact Riemann surface.

If $p\equiv 1$ $\left( \mathrm{mod}4\right) ,$ then $\Gamma \left(
A,p\right) $ has no elliptic elements.

Let $S\left( t\right) =S^{+}\left( t\right) -S^{-}\left( t\right) ,$ where
S^{+}\left( t\right) =\max \left\{ 0,S\left( t\right) \right\} $ and
S^{-}\left( t\right) =\max \left\{ 0,-S\left( t\right) \right\} .$ Then the
following theorem holds.

\begin{theorem}[Hejhal-Selberg]
Let $\Gamma =\Gamma \left( A,p\right) $ with $p\equiv 1$ $\left( \mathrm{mod
4\right) .$ Then (for large $T$):
\begin{equation*}
\frac{1}{T}\int_{T}^{qT}S^{+}\left( t\right) ^{2}dt\geq c_{1}\frac{T}{\left(
\log T\right) ^{2}}
\end{equation*
where $c_{1}$ is a positive constant that depends only on $\Gamma .$ A
similar inequality holds for $S^{-}\left( t\right) .$
\end{theorem}

In order to appreciate this result note that it implies that the average
deviation of $S\left( T\right) $ from its mean (if it exists) is of the
order larger than $\sqrt{T}/\left( \log T\right) $ which should be compared
with the number of zeros in the interval $\left[ 0,T\right] ,$ that is,
cT^{2}.$ To put it in prospective note that the average deviation of the
zeros of $S\left( T\right) $ for Riemann's zeta function is of the order
\left( \log \log T\right) ^{1/2}$ which should be compared with $cT\log T,$
the number of Riemann's zeros in $\left[ 0,T\right] .$ These situations
appear to be quite different.

Moreover, recently there was some numeric and theoretical work on the
eigenvalues of the Laplace operator on manifolds $\Gamma \backslash \mathbb{
}$ for arithmetic groups $\Gamma .$ First, numeric and heuristic analysis
showed that the spacings between eigenvalues resemble spacings between
points from a Poisson point process rather than spacings between eigenvalues
of a random matrix ensemble (see Bogomolny et al \cite{bggs92} and
references wherein). Next some rigorous explanations of this finding have
been given. They are based on exceptionally large multiplicities of closed
geodesics with the same length. See Luo and Sarnak (\cite{luo_sarnak94a} and
\cite{luo_sarnak94b}) and Bogomolny et al \cite{bogomolny_leyvraz_schmit96}.

There is also some work on correlations of closed geodesics - see Pollicott
and Sharp \cite{pollicott_sharp06}.

\subsection{Comparison with the circle problem}

If we consider the compact Riemann surface of genus $1,$ then we are led to
similar questions. Such a surface can be represented as quotient space
\Lambda \backslash \mathbb{C}$, where $\Lambda $ is a lattice. Consider, for
concreteness, $\Lambda =\left[ 1,i\right] .$ Then the eigenvalues of the
Laplace operator are $4\pi ^{2}\left( m^{2}+n^{2}\right) ,$ where $m$ and $n$
are integer, and the number of the eigenvalues below $t$ equals the number
of integer points in the circle $t/\pi .$ Let
\begin{equation*}
r\left( n\right) =N\left\{ \left( a,b\right) \in \mathbb{Z}\times \mathbb{Z
:a^{2}+b^{2}=n\right\} ,
\end{equation*
and
\begin{equation*}
A\left( x\right) =\sum_{0\leq n\leq x}r\left( n\right) =\pi x+R\left(
x\right) .
\end{equation*}

The function $A\left( x\right) $ can be thought as the counting function
both for eigenvalues of Laplace operator and for closed geodesics of bounded
length.

Then by using the Poisson summation formula it is possible to derive the
following result:
\begin{equation*}
\sum r\left( n\right) f\left( n\right) =\pi \sum r\left( n\right)
\int_{0}^{\infty }f\left( x\right) J_{0}\left( 2\pi \sqrt{nx}\right) dx.
\end{equation*
Given that it is possible to apply this to the step function $f\left(
x\right) ,$ one can obtain the explicit formula:
\begin{equation*}
R\left( x\right) =\sqrt{x}\sum_{n=1}^{\infty }\frac{r\left( n\right) }{\sqrt
n}}J_{1}\left( 2\pi \sqrt{nx}\right) .
\end{equation*
This leads to various estimates on $R\left( x\right) ,$ in particular it is
known tha
\begin{equation*}
R=O\left( x^{1/3}\right) \text{ and }R=\Omega _{\pm }\left( x^{1/4}\right) ,
\end{equation*
and tha
\begin{equation*}
\frac{1}{x}\int_{0}^{x}R\left( t\right) ^{2}dt=cx^{1/2}+O\left[ \left( \log
x\right) ^{3}\right] .
\end{equation*}

This suggest that the \textquotedblright standard
deviation\textquotedblright\ of $R\left( t\right) $ is $x^{1/4}.$ Again this
does not resemble the situation with Riemann's zeta zeros.

Some more details about this problem and references to the early papers can
be found in \cite{kendall48}. More recent research include \cit
{heath-brown92} and \cite{bcdl93}.

\section{Zeta functions of dynamical systems}

The main sources for this section are reviews by Ruelle (\cite{ruelle92} and
\cite{ruelle02}) and Pollicott (\cite{pollicott03} and \cite{pollicott10}).

\subsection{Zetas for maps}

Let $f$ be a map of a set $M$ to itself, let the periodic orbits of $f$ be
denoted by $P,$ and let $\left\vert P\right\vert $ denote the period of
orbit $P.$ Then, we can define the zeta of $f$ by the following formula:
\begin{eqnarray*}
\zeta \left( z\right) &=&\prod_{P}\left( 1-z^{\left\vert P\right\vert
}\right) ^{-1} \\
&=&\exp \sum_{m=1}^{\infty }\frac{z^{m}}{m}\left\vert \mathrm{Fix}\text{
f^{m}\right\vert ,
\end{eqnarray*
where $\left\vert \mathrm{Fix}\text{ }f^{m}\right\vert $ denote the number
of fixed points of $f^{m}.$

\subsubsection{Permutations}

Let $M$ be a finite set, and let $f$ be given by a permutation matrix $A.$
Then the number of fixed points of $f^{m}$ is given by \textrm{Tr}$A^{m}.$
Hence, we have
\begin{eqnarray*}
\zeta \left( z\right) &=&\exp \mathrm{Tr}\sum_{m=1}^{\infty }\frac{\left(
zA\right) ^{m}}{m} \\
&=&\exp \left( -\mathrm{Tr\log }\left( 1-zA\right) \right) \\
&=&1/\det \left( 1-zA\right) ,
\end{eqnarray*
which is closely related to the characteristic polynomial of matrix $A.$ In
particular, all poles of the zeta are on the unit circle.

\subsubsection{Diffeomorphisms}

Let $M$ be a torus $\mathbb{R}^{2}/\mathbb{Z}^{2}$ and let $f$ be induced by
a linear transformation $A\in SL_{2}\left( \mathbb{Z}\right) .$ Assume that
the eigenvalues of $A$ are positive and not on the unit circle: $\lambda
_{1}>1>\lambda _{2}>0.$ Then the number of the fixed points of $f^{m}$ is
the number of solutions of the equation $I-A^{m}=0$ $\left( \mathrm{mod
\mathbb{Z}^{2}\right) ,$ which can be computed as $\left\vert \mathrm{\det
\left( I-A^{m}\right) \right\vert =\left( \lambda _{1}^{m}-1\right) \left(
1-\lambda _{2}^{m}\right) .$ Hence, we have
\begin{eqnarray*}
\zeta \left( z\right) &=&\exp \mathrm{Tr}\sum_{m=1}^{\infty }\frac{1}{m
\left[ \left( z\lambda _{1}\right) ^{m}+\left( z\lambda _{2}\right)
^{m}-z^{m}-\left( z\lambda _{1}\lambda _{2}\right) ^{m}\right] \\
&=&\frac{\left( 1-z\lambda _{1}\right) \left( 1-z\lambda _{2}\right) }
\left( 1-z\right) \left( 1-z\lambda _{1}\lambda _{2}\right) } \\
&=&\frac{\left( 1-z\lambda _{1}\right) \left( 1-z\lambda _{2}\right) }
\left( 1-z\right) ^{2}}.
\end{eqnarray*
This example can be generalized to linear transformations $A\in GL_{d}\left(
\mathbb{Z}\right) $ acting on a $d$-dimensional torus or more generally to
discrete groups acting on compact quotients of nilpotent Lie groups
(nil-manifods).

If we wish to generalize this example to diffeomorphisms of smooth
orientable manifolds, it is natural to count the fixed points by taking into
account the degree of map $f$ at the fixed points. Recall that the degree
equals to $1$ if the map is non-singular and preserves orientation at the
fixed point, and to $-1$ if it reverses the orientation. It can also be
generalized to the situation when the map is singular at the fixed point. If
$d\left( x,f\right) $ denotes the degree of $f$ at a fixed point $x,$ then
we define the Lefschetz zeta function as
\begin{equation*}
\zeta _{L}\left( z\right) =\exp \sum_{m=1}^{\infty }\frac{z^{m}}{m
\sum_{x\in \mathrm{Fix}\left( f^{m}\right) }d\left( x,f^{m}\right) .
\end{equation*
In this case one can use the Lefschetz fixed point formula that says:
\begin{equation*}
\sum_{x\in \mathrm{Fix}\left( f^{m}\right) }d\left( x,f^{m}\right)
=\sum_{i=0}^{\dim M}\left( -1\right) ^{i}\mathrm{Tr}\left( \left(
f^{m}\right) _{\ast i}:H_{i}\rightarrow H_{i}\right) ,
\end{equation*
where $H_{i}$ is the $i$-th homology group of the compact manifold $M$ with
real coefficients, and $\left( f^{m}\right) _{\ast i}$ is the map induced by
$f^{m}$ on $H_{i}.$

In particular, if $\lambda _{ij}$ are eigenvalues of $f_{\ast i},$ then we
get
\begin{eqnarray*}
\zeta _{L}\left( z\right) &=&\exp \sum_{m=1}^{\infty }\frac{1}{m
\sum_{i=0}^{\dim M}\left( -1\right) ^{i}\sum_{j=1}^{\dim H_{i}}\left(
z\lambda _{ij}\right) \\
&=&\prod_{i=0}^{\dim M}\left( \prod_{j=1}^{\dim H_{i}}\left( 1-z\lambda
_{ij}\right) ^{-1}\right) ^{\left( -1\right) ^{i}},
\end{eqnarray*
which is a rational function.

If $f$ is a diffeomorphism of a compact manifold $M,$ then the original
dynamical zeta (counted without taking into account the degree of $f^{m}$)
is often called the \emph{Artin-Mazur zeta function} (see Artin-Mazur \cit
{artin_mazur65}). If the diffeomorphism satisfies some additional conditions
(hyperbolicity or Axiom A), then it is known that this function is rational.
(This was conjectured by Smale \cite{smale67}, and proved by Guckenheimer
\cite{guckenheimer70} and Manning \cite{manning71}.)

\subsubsection{Subshifts}

Suppose next that $A$ is an $N$-by-$N$ matrix of zeros and ones, and that
the set $M$ consists of doubly infinite sequences $\left\{ x_{i}\right\} $
of symbols $1,\ldots ,N,$ which satisfy the following criterion. A sequence
\left\{ x_{i}\right\} $ belongs to $M$ if and only if $A_{x_{i}x_{i+1}}=1$
for every $i.$ In other words, the symbol $x_{i}$ determines which of the
other symbols are possible candidates for $x_{i+1}.$ The map $f$ is simply a
shift on this set $M$: $\left\{ x_{i}\right\} \rightarrow \left\{
x_{i+1}\right\} .$ In this case, the number of fixed points of $f^{m}$ is
simply $\mathrm{Tr}\left( A^{m}\right) ,$ and we have
\begin{equation}
\xi \left( z\right) =1/\det \left( 1-zA\right) ,
\label{formula_det_subshift}
\end{equation
similar to Example 1. Note that if $A$ is symmetric then the dynamics of the
system is in a certain sense reversible. On the other hand the poles of the
zeta in this case are all real. So this example illustrate a connection
between the distribution of zeros and the reversibility of a system.

\subsubsection{Ihara-Selberg zeta function}

A basic reference for this section is a book by Terras \cite{terras11}.

Let $G$ be a finite graph. Orient its edges arbitrarily. Let the
2\left\vert E\right\vert $ oriented edges be denoted $e_{1},e_{2},\ldots
,e_{n},e_{n+1}=e_{1}^{-1},\ldots ,e_{2n}=e_{n}^{-1}.$ A \emph{path} is a
sequence of oriented edges such that the end of one edge equals to the
beginning of the next edge. A path is \emph{closed} if the end of the last
edge corresponds to the beginning of the first edge. A closed path $\left(
e_{1},e_{2},\ldots ,e_{s}\right) $ is \emph{equivalent} to the path $\left(
e_{2},\ldots ,e_{s},e_{1}\right) .$ A closed path $P$ is \emph{primitive} if
$P\neq D^{m}$ for $m\geq 2$ and any other path $D.$ A path is \emph
non-backtracking} if $e_{i+1}\neq e_{i}^{-1}$ for any $i.$ A closed path is
\emph{tailless} if $a_{s}\neq a_{1}^{-1}.$

If $P=\left( e_{1},e_{2},\ldots ,e_{s}\right) ,$ then $l\left( P\right) =s$
is the length of the path.

A prime $\left[ P\right] $ is an equivalence class of primitive closed
non-backtracking tailless paths. (It is an analogue of a geodesic on a
manifold.)

The \emph{Ihara zeta function} is defined as follows
\begin{equation*}
\zeta _{G}\left( u\right) =\prod_{\left[ P\right] }\left( 1-u^{l\left(
P\right) }\right) ^{-1}.
\end{equation*}

It has a representation as a determinant which was first discovered by Ihara
for regular graphs (\cite{ihara66}) and then by Bass \cite{bass92} and
Hashimoto \cite{hashimoto89} for arbitrary finite graphs:
\begin{equation*}
\zeta _{G}\left( u\right) ^{-1}=\left( 1-u^{2}\right) ^{\left| E\right|
-\left| V\right| }\det \left( I-A_{G}u+Q_{G}u^{2}\right) ,
\end{equation*
where $A_{G}$ is the adjacency matrix of $G,$ and $Q_{G}$ is the diagonal
matrix whose $j$-th diagonal entry is ($-1$ + degree of $j$-th vertex).

The Ihara zeta function can be though as a particular case of a dynamical
zeta function for a subshift associated with graph $G.$ The alphabet
consists of $2\left| E\right| $ oriented edges, and the subshift matrix
equals to the edge adjacency matrix $W_{G}$ which is defined as follows.

\begin{definition}
The \emph{edge adjacency matrix} $W_{G}$ is the $2\left| E\right| $-by-
2\left| E\right| $ such that $i,j$ entry equals $1$ if the terminal vertex
of edge $i$ equals the initial vertex of edge $j$ and edge $j$ is not the
inverse of edge $i.$
\end{definition}

In particular, from (\ref{formula_det_subshift}) we have another
determinantal formula
\begin{equation*}
\zeta _{G}\left( u\right) ^{-1}=\det \left( I-uW_{G}\right) .
\end{equation*}

If the graph is $q+1$ regular, that is, if every vertex has degree $q+1,$
then there is a pole of the zeta at $u=1/q$. Moreover, every non-real pole
of the zeta lies on the circle with radius $1/\sqrt{q}.$ The Riemann
hypothesis for regular graphs says that all non-trivial poles are on this
circle. This is not always true, and a useful criterion is as follows. Let
the second largest in magnitude eigenvalue of $A_{G}$ be denoted $\lambda
_{1}.$ Then the Riemann hypothesis holds if and only if $\left\vert \lambda
_{1}\right\vert \leq 2\sqrt{q}.$ The graphs that satisfy this condition are
often called Ramanujan following a paper by Lubotsky, Phillips, and Sarnak
\cite{lubotzky_phillips_sarnak88}, which constructed an infinite family of
such graphs.

Note that $2\sqrt{q}$ is critical in the sense that for every $\varepsilon
>0,$ there exists a constant $C$ such that for all $(q+1)$-regular graphs
with $n$ vertices there is at least $Cn$ eigenvalues in the interval $\left[
2\sqrt{q}-\varepsilon ,q+1\right] .$ On the other hand, a random regular
graph is approximately Ramanujan with high probability. That is, for
arbitrary $\varepsilon >0,$ the probability that $\left| \lambda _{1}\right|
\geq 2\sqrt{q}+\varepsilon $ becomes arbitrarily small as the size of the
graph grows. This fact is known as the Alon conjecture and it was recently
proved by J. Friedman in Friedman (2008). However, the distribution
properties of the largest eigenvalue are still unknown.

In his Ph.D. thesis \cite{newland05}, Derek Newland studied spacings of
eigenvalues of random regular graphs and spacings of the Ihara zeta zeros
and found numerically that they resemble spacings in the Gaussian Orthogonal
Ensemble. See also an earlier paper by Jacobson, Miller, Rivin and Rudnick
\cite{jmrr99}.

For Ihara'z zeta function, there is an analog of Selberg's trace formula
which we formulate for the case of\ a $q+1$ regular graph.

Let $N_{m}$ be the number of all non-backtracking tailless closed paths of
length $m.$ (Note that this is the number of all closed paths, not the
number of the equivalency classes.) In particular,
\begin{eqnarray*}
u\frac{d}{du}\log \zeta _{G}\left( u,X\right) &=&\sum_{\left[ P\right]
}l\left( P\right) \left( u^{l\left( P\right) }+u^{2l\left( P\right)
}+...\right) \\
&=&\sum_{P}\left( u^{l\left( P\right) }+u^{2l\left( P\right) }+...\right) \\
&=&\sum N_{m}u^{m}.
\end{eqnarray*}

This shows tha
\begin{equation*}
\zeta _{G}\left( u\right) =\exp \sum_{m\geq 1}N_{m}\frac{u^{m}}{m}.
\end{equation*}

\begin{theorem}[Terras]
Suppose $0<a<1/q$. Assume that $h\left( u\right) $ is meromorphic in the
plane and holomorphic for $\left\vert u\right\vert >a-\varepsilon ,$
\varepsilon >0$ Assume that \ $h\left( u\right) =O\left( \left\vert
u\right\vert ^{-1-\alpha }\right) ,$ $\alpha >0.$ Let $\widehat{h}\left(
n\right) :=\frac{1}{2\pi i}\int_{\left\vert u\right\vert =a}u^{n}h\left(
u\right) du$ and assume that $\widehat{h}\left( n\right) $ decays rapidly
enough. Then,
\begin{equation*}
\sum_{\rho }\rho h\left( \rho \right) =\sum_{n\geq 1}N_{n}\widehat{h}\left(
n\right) ,
\end{equation*
where the sum on the left is over the poles of $\zeta _{G}\left( u\right) .$
\end{theorem}

Some applications of this result can be found in ...

\subsubsection{Frobenius maps}

Another important example arises if $M$ is the set of solutions of a system
of algebraic equations over the algebraic closure of $\mathbb{F}_{q}$ (i.e.,
the finite field with $q=p^{n}$ elements), and $f$ is the Frobenius map
Frob $: $x\rightarrow x^{q}.$ These functions are often called the \emph
Weil zeta functions} although it should be noted that the first examples
were originally considered by E. Artin.

Weil's zeta functions directly connected to the number-theoretical
zeta-functions. Consider the affine curve given by the equation $f\left(
X,Y\right) =0$ over the finite field $\mathbb{F}_{q}.$ Let $\mathfrak{p}$
denote a prime ideal of the field $\mathbb{F}_{q}\left[ X,Y\right] /f\left(
X,Y\right) $ and let the order of $\mathbb{F}_{q}\left[ X,Y\right]
\mathfrak{p}$ be denoted by $N\mathfrak{p.}$ Then, by analogy with Riemann's
zeta function we can define
\begin{equation*}
\zeta \left( s\right) =\prod_{\mathfrak{p}}\frac{1}{1-N\mathfrak{p}^{-s}}.
\end{equation*
It turns out that this definition is in agreement with another definition of
the Weil zeta function
\begin{equation*}
\zeta \left( s\right) =\exp \sum_{m=1}^{\infty }N_{m}\frac{p^{-sm}}{m},
\end{equation*
where $N_{m}$ is the number of fixed points of $Frob^{m},$ or to put it in
other words, the number of solutions of the equation $f\left( X,Y\right) =0$
in the finite field $\mathbb{F}_{q^{m}}.$

Here is an example. Consider the affine line. Then $N_{m}=q^{m},$ and
\begin{eqnarray*}
\zeta \left( z\right)  &=&\exp \sum_{m=1}^{\infty }N_{m}\frac{z^{m}}{m} \\
&=&\exp \left[ -\log \left( 1-qz\right) \right]  \\
&=&\frac{1}{1-qz}.
\end{eqnarray*
In another example, the curve is elliptic. Note that the Frobenius map is a
linear operator over $\mathbb{F}_{q}$ and there is some resemblance to
Example 2 about linear maps on a torus. We can guess then that
\begin{equation*}
\zeta \left( z\right) =\frac{1-z\mathrm{Tr}\left[ Frob\right] +z^{2}\det
\left[ Frob\right] }{\left( 1-z\right) \left( 1-z\det \left[ Frob\right]
\right) },
\end{equation*
provided $\mathrm{Tr}\left[ Frob\right] $ and $\det \left[ Frob\right] $ are
given some appropriate meaning. If we believe $\det \left[ Frob\right] =q,$
then it $N_{1}=\det \left( I-Frob\right) =\det (Frob)-\mathrm{Tr}\left(
Frob\right) +1,$ and so $\mathrm{Tr}\left[ Frob\right] =q+1-N_{1}.$ Hence,
our guess would be
\begin{equation*}
\zeta \left( z\right) =\frac{1-z\left( q+1-N_{1}\right) +z^{2}q}{\left(
1-z\right) \left( 1-zq\right) },
\end{equation*
and it turns out that this guess is correct. Of course, $M$ is in fact
infinite-dimensional over $\mathbb{F}_{q},$ so we do not know what is the
meaning of the determinant and the trace. However, one of the proofs of the
rationality of the zeta function uses a two-dimensional module over $\mathbb
Q}_{l}$ (the field of $l$-adic numbers) and the arguments resemble our
heuristic. One can consult the book by Silverman \cite{silverman86} to see
more. A more standard proof is based on the Riemann-Roch theorem and
counting divisors. It can be found in \cite{milne06}.

The Riemann hypothesis in this example is equivalent to the statement that
\begin{equation*}
\left| N_{1}-q-1\right| \leq 2\sqrt{q},
\end{equation*
since this implies that the roots of the polynomial in the numerator are on
the circle with radius $q^{-1/2}.$

Artin \cite{artin24} proved the zetas of the curves can be explicitly
computed as rational functions of $z=p^{-s}.$ He checked numerically that in
many cases the zeta function satisfies an appropriate variant of the Riemann
hypothesis, and conjectured that this should be true for all curves. Later
this was proved by Hasse \cite{hasse36} and Weil (\cite{weil40} and \cit
{weil41}).

Weil also conjectured in \cite{weil49} that the zeta function of every
algebraic variety is rational, that it has a functional equation, and that
it satisfies the Riemann hypothesis. The proof of this general conjecture
led to an introduction of many new ideas in algebraic geometry by Dwork,
Grothendick, and Deligne.

Katz and Sarnak \cite{katz_sarnak99} have studied the distribution of the
zeros of the Weil zeta functions when the genus of the corresponding curve
grows to infinity. They found that for \textquotedblleft
most\textquotedblright\ of the curves the local statistical behavior of
these zeros approaches the behavior of the eigenvalues from the random
matrix ensembles. It is interesting that there are no explicit constructions
of curve families for which it is known to be true. It is only known that it
must be true for most of the curves.

\subsubsection{Maps of an interval}

For yet another example consider the map $x\rightarrow 1-\mu x^{2}$ of the
interval $\left[ -1,1\right] $ to itself. For a special value of $\mu
\approx 1.401155...$ (the Feingenbaum value), this map has one periodic
orbit of period $2^{n}$ for every integer $n\geq 0.$ Therefore,
\begin{equation*}
\zeta \left( z\right) =\prod_{n=0}^{\infty }\left( 1-z^{2^{n}}\right)
^{-1}=\prod_{n=0}^{\infty }\left( 1+z^{2^{n}}\right) ^{n+1}.
\end{equation*
This $\zeta $ satisfies the functional equation $\zeta \left( z^{2}\right)
=\left( 1-z\right) \zeta \left( z\right) .$ More generally, the piecewise
monotone maps of the interval $\left[ -1,1\right] $ to itself correspond to
the \emph{Milnor-Thurston zeta functions}. See \cite{milnor_thurston88}. As
far as I know there was no systematic study of statistical properties of
their poles and zeros.

\subsection{Zetas for flows}

If $f$ is a flow on $M,$ that is, a map $M\times \mathbb{R}^{+}\rightarrow
M, $ then we can define the zeta function of this flow as
\begin{equation*}
\zeta \left( s\right) =\prod_{\omega }\left( 1-e^{-sl\left( \omega \right)
}\right) ^{-1},
\end{equation*
where $\omega $ denotes a periodic orbit of $f,$ and $l\left( \omega \right)
$ is its length. It is a more general case than the case of maps since there
is a construction (\textquotedblleft suspension\textquotedblright ) that
allows us to realize maps as flows but not vice versa. Unsurprisingly, it
turns out that zeta functions for flows are more difficult to investigate
than zeta functions for maps.

If we imagine that prime numbers correspond to periodic orbits of a flow and
that the length of the orbit $p$ is given by $\log p,$ then the zeta
function of the flow will coincide with Riemann's zeta function.

One particularly important example of a flow is the geodesic flow on a
smooth manifold $M.$ In the case when $M$ has a constant negative curvature,
the corresponding dynamical zeta function is closely related to the Selberg
zeta function. Namely,
\begin{equation*}
\zeta \left( s\right) =\frac{Z\left( s+1\right) }{Z\left( s\right) }\text{
and }Z\left( s\right) =\prod_{n=0}^{\infty }\zeta \left( s+n\right) ^{-1}.
\end{equation*
Another important example is the geodesic flow for billiards on polygons.

I am not aware about the functional equation for zeta functions that comes
from general geodesic flows (other than flows on manifolds of a constant
negative curvature). \ Also, it appears that not much is known about the
statistical properties of the distribution of zeros and poles of these zeta
functions.

The main benefit of these functions is that the distribution of closed
geodesics depends on the maximal real part of their poles, and this quantity
can be studied by methods that do not involve the functional equation or
trace formula. Hence, one can study geodesics on spaces of variable
curvature. The idea of this approach is that geodesic flows can be written
as shifts of sequences of symbols and poles of zeta functions are related to
eigenvalues of operators induced by these shifts. (This is the method of
symbolic dynamics).

\section{Conclusion}

We considered the statistical properties of the zeta functions. In summary,
the zeros of number-theoretical zeta functions satisfies many properties
which are true for eigenvalues of random matrices. The main outstanding
problem is the Montgomery conjecture that says that the correlations are the
same as for random matrices even if the test functions are not smooth.

The statistical properties of the Selberg zeta zeros are different from the
properties of the eigenvalues of random matrices. It is conjectured that the
local properties coincide with the properties of the random independent
points However, the exact description of these properties is not known.

The zeros of dynamical zeta functions are often related to eigenvalues of
certain matrices, and the study of their properties depends on the study of
the matrix properties. Two interesting cases are the case of zeta functions
of algebraic curves and the case of Ihara's zeta functions of graphs.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


Drill down on organization code instead of account and you can review multiple accounts (up to 10 at a time) on a single screen by clicking on button at bottom of screen to acknowledge review.
Make sure you have a proper Separation of Duties — review the attached matrices (XLS) for suggestions on how your department can accomplish an appropriate Separation of Duties.
Can you consolidate or otherwise reduce/inactivate some accounts that you manage?
Can some of your accounts be re-assigned to a staff member in your departmental Dean's Office or to one of your account delegates? For example, if you are listed as the fiscal officer for 650 accounts, can you re-assign one of your account delegates to be the fiscal officer on 300 of those accounts?
Make sure all of your accounts have an active account delegate. Every FIS account should have a delegate for ALL KFS document types that is not the same as the fiscal officer. We recommend that you schedule the Account Delegate Lookup (167) in FIS Decision Support (DS) so that you can regularly ensure that all of your accounts have a delegate.
Most department-initiated KFS documents involving financial transactions (excluding Budget Adjustment and Pre-Encumbrance documents) will require a separate initiator and approver. In other words, if a Fiscal Officer initiates a document, KFS will require one of their delegates to approve that document. For documents initiated by an Account Delegate, the Fiscal Officer or another Delegate would need to approve the document to satisfy separation of duties. If a Fiscal Officer initiates a document for which there is no account delegate, they will receive a warning message advising them that a Delegate must be assigned to the account in order to submit the document. They will not be able to submit the document until a delegate has been assigned to the account.
Do not assign fiscal officers as a Primary Preparer or Mandatory Reviewer in DRS. Fiscal officers should not be assigned as a Primary Preparer or Mandatory Reviewer in the PPS/OPTRS Departmental Roles Structure (DRS) for any of their assigned accounts.
As departments are updating their PPS/OPTRS Departmental Roles Structures (DRS), note that Back-up PPS/OPTRS preparers who are also fiscal officers shouldn't prepare more than 10% of departmental PPS/OPTRS transactions.


Lankes; R.D. & Radford; M.L.
Building on an orientation program established by the University of Nevada, Las Vegas (UNLV) Libraries' Human Resources unit for all new employees, the Research and Information Department developed a training and orientation program for new employees who participate in the reference desk service pool. A task force composed of service pool members created a checklist of expectations that led to a new mission statement and core values for the department as well as a time line for training new employees. The use of a staff wiki to provide access to the expectations, time line, and other resources is explored. Assessment mechanisms, adjustments to the program, and future plans are detailed.
Lowe, S. D., Nozero, V. (2010). Utilizing new staff training initiatives to develop and implement reference competencies at the UNLV Libraries. In Lankes; R.D. & Radford; M.L., Reference Renaissance: Current and Future Trends 279-286. Chicago, IL: Neal-Schuman Publishers.