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\begin{document}

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\section*{zeta function}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{theta_functions}{}\paragraph*{{Theta functions}}\label{theta_functions}

[[!include theta functions - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{FunctionFieldAnalogy}{Function field analogy}\dotfill \pageref*{FunctionFieldAnalogy} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_algebraic_geometry}{In algebraic geometry}\dotfill \pageref*{in_algebraic_geometry} \linebreak
\noindent\hyperlink{categorical_approaches}{Categorical approaches}\dotfill \pageref*{categorical_approaches} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The concept of \emph{zeta function} originates in [[number theory]], but to get an idea of what they ``really are'' it is helpful to proceed anachronistically:

$\zeta$-functions are [[meromorphic functions]] $s \mapsto \zeta(s)$ on the [[complex plane]], which behave like like [[analytic continuations]] of [[traces]] of powers

\begin{displaymath}
s \mapsto Tr \left(\frac{1}{H}\right)^s
\end{displaymath}
of suitable [[elliptic differential operators]] $H$ (in [[physics]] these are [[regularization (physics)|regularized]] [[traces]] of [[Feynman propagators]] leading to expressions for [[vacuum amplitudes]]), which means that for sufficiently nice such $H$ these are analytic continuations in $s$ of sums of the form

\begin{displaymath}
s \mapsto \underset{\lambda}{\sum}  \lambda^{-s}
  \,,
\end{displaymath}
where the summation is over the [[eigenvalues]] $\lambda$ of $H$.

Indeed, such \emph{[[zeta functions of elliptic differential operators]]} constitutes one class of examples of zeta functions. Of particular interest is the case where $H$ is a [[Laplace operator]] of a [[hyperbolic manifold]] and in particular on a hyperbolic [[Riemann surface]], for that case one obtains the \emph{[[zeta function of a Riemann surface]]}, in particular the \emph{[[Selberg zeta function]]}.

In modern language one also speaks of \emph{[[L-functions]]}. Where a zeta function of some space is like the [[Feynman propagator]] of \emph{the} canonical [[Laplace operator]] of that space, an L-function is defined from an extra ``twisting'' information such as that of a [[flat bundle]]/[[local system of coefficients]] on the space (and is hence like the [[Feynman propagator]] of the corresponding twisted/coupled [[Laplace operator]]). The major properties satisfied by anything that qualifies as a zeta function or [[L-functions]] are: these are [[meromorphic functions]] $s \mapsto L(s)$ on the [[complex plane]] such that

\begin{enumerate}%
\item for $\Re(s) \gt 1$ they have a [[convergence|converging]] [[series]] expansion of the above form, and/or a [[infinite product|multiplicative series]] expression, the \emph{[[Euler product]]};


\item such that [[analytic continuation]] of the series expression exists to a meromorphic function $L(-)$ on the complex plane;


\item and such that the result satisfies a \emph{[[functional equation]]} which says that the product $\hat L$ of $L$ with some correcion functions satisfies $\hat L(1-s) = \hat L(s)$.



\end{enumerate}
Proceeding from the above class of examples in [[complex analytic geometry]] one may wonder if there are [[analogy|analogs]] also in [[arithmetic geometry]]. Indeed, by the [[function field analogy]] there are. All the way down ``on [[Spec(Z)]]'' the analog of the [[Selberg zeta function]] is the [[Riemann zeta function]], which \emph{historically} is the first of all zeta functions, defined by [[analytic continuation]] of the [[series]]

\begin{displaymath}
s \mapsto \underoverset{n = 1}{\infty}{\sum} n^{-s}
  \,.
\end{displaymath}
The \emph{[[Riemann hypothesis]]} [[conjecture|conjectures]] a characterization of the [[roots]] of this zeta function and is regarded as one of the outstanding problems in [[mathematics]]. It has evident analogs for all other zeta functions (for some of which it has been proven).

More generally, over [[arithmetic curves]] which are [[spectrum of a commutative ring|spectra]] of [[rings of integers]] of more general [[number fields]], the Riemann zeta function has generalization to the \emph{[[Artin L-functions]]} defined intrinsically in terms of [[characteristic polynomials]] of [[Galois representations]]. When the Galois representation is 1-dimensional, then the Artin L-function may be expressed (by ``[[Artin reciprocity]]'') in terms of ``more arithmetic'' data by [[Dirichlet L-functions]] and [[Hecke L-functions]]. When the Galois representation is higher dimensional, then the [[Langlands correspondence]] [[conjecture]] asserts that the Artin L-function may be expressed ``arithmetically'' as the [[automorphic L-function]] of an [[automorphic form]].

Similarly on [[arithmetic curves]] given by [[function fields]] there is the [[Goss zeta function]] and in [[higher dimensional arithmetic geometry]] the [[Weil zeta function]], famous from the \emph{[[Weil conjectures]]}. The

When interpreting the [[Frobenius morphisms]] that appear in the [[Artin L-functions]] geometrically as flows (as discussed at \emph{\href{Borger's+absolute+geometry#Motivation}{Borger's arithmetic geometry -- Motivation}}) then this induces an evident analog of [[zeta function of a dynamical system]]. This in turn has strong analogies with [[Alexander polynomials]] in [[knot theory]] (see at \emph{[[arithmetic topology]]}).

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{FunctionFieldAnalogy}{}\subsubsection*{{Function field analogy}}\label{FunctionFieldAnalogy}

One way to understand the plethora of different zeta functions is to see them as the incarnation of the same general concept in different flavors of [[geometry]]. This is expressed at least in parts by the

[[!include function field analogy -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[multiple zeta values]], [[motivic multiple zeta values]], [[motivic integration]], [[motive]]


\item [[Weil conjecture]]


\item [[Riemann hypothesis]]


\item there are attempts to understand the Riemann zeta function as the spectrum of a [[Hamiltonian]] of a [[quantum mechanical system]]. See at \emph{[[Riemann hypothesis and physics]]}.


\item [[Beilinson regulator]]


\item [[zeta function of a Riemann surface]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

A useful survey of the zoo of zeta functions is in

\begin{itemize}%
\item [[Jeffrey Lagarias]], \emph{Number theory zeta functions and Dynamical zeta functions} (\href{http://www.math.lsa.umich.edu/~lagarias/doc/numberthzeta.pdf}{pdf})

\end{itemize}
Further general review includes

\begin{itemize}%
\item E. Kowalski, first part of \emph{Automorphic forms, L-functions and number theory (March 12--16) Three Introductory lectures} (\href{http://www.math.ethz.ch/~kowalski/lectures.pdf}{pdf})


\item [[Alain Connes]], [[Matilde Marcolli]], chapter II of \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]}



\end{itemize}
Discussion in the more general context of [[higher dimensional arithmetic geometry]] is in

\begin{itemize}%
\item [[Ivan Fesenko]], \emph{Adelic approch to the zeta function of arithmetic schemes in dimension two}, Moscow Math. J. 8 (2008), 273--317 (\href{https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf}{pdf})

\end{itemize}
\hypertarget{in_algebraic_geometry}{}\subsubsection*{{In algebraic geometry}}\label{in_algebraic_geometry}

\begin{itemize}%
\item [[Yuri Manin]], \emph{Lectures on zeta functions and motives (according to Deninger and Kurokawa)}, Ast\'e{}risque \textbf{228}:4 (1995) 121--163, and preprint MPIM1992-50 \href{http://www.mpim-bonn.mpg.de/preblob/4793}{pdf}


\item Nobushige Kurokawa, \emph{Zeta functions over $F_1$}, Proc. Japan Acad. Ser. A Math. Sci. \textbf{81}:10 (2005) 180-184 \href{http://projecteuclid.org/euclid.pja/1135791771}{euclid}


\item [[Bruno Kahn]], \emph{Fonctions z\^e{}ta et $L$ de vari\'e{}t\'e{}s et de motifs}, \href{http://arxiv.org/abs/1512.09250v1}{arXiv:1512.09250}.



\end{itemize}
\hypertarget{categorical_approaches}{}\subsubsection*{{Categorical approaches}}\label{categorical_approaches}

\begin{itemize}%
\item [[Michael Larsen|M. Larsen]], [[Valery Lunts|V. A. Lunts]], \emph{Motivic measures and stable birational geometry}, Mosc. Math. J. \textbf{3}, 1 (2003) 85--95; \emph{Rationality criteria for motivic zeta functions}, Compos. Math. \textbf{140}:6 (2004) 1537--1560


\item [[Vladimir Guletskii]], \emph{Zeta functions in triangulated categories}, Mathematical Notes \textbf{87}, 3 (2010) 369--381, \href{http://arxiv.org/abs/math/0605040}{math/0605040}


\item [[Maxim Kontsevich|M. Kontsevich]], \emph{Notes on motives in finite characteristics}, \href{http://arxiv.org/abs/math.AG/0702206}{math.AG/0702206}


\item [[John Baez]], \emph{[[johnbaez:Zeta functions]]}


\item [[Sergey Galkin]], [[Evgeny Shinder]], \emph{On a zeta-function of a dg-category}, \href{http://arxiv.org/abs/1506.05831}{arXiv:1506.05831}.



\end{itemize}
[[!redirects zeta functions]]



\end{document}