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\section*{Selberg zeta function}

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

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

[[!include theta functions - contents]]

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

\begin{quote}%
under construction


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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_evendimensional_manifolds}{For even-dimensional manifolds}\dotfill \pageref*{for_evendimensional_manifolds} \linebreak
\noindent\hyperlink{DefinitionForOdd}{For odd-dimensional manifolds}\dotfill \pageref*{DefinitionForOdd} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AnalogyWithArtinLFunction}{Analogy with Artin L-function}\dotfill \pageref*{AnalogyWithArtinLFunction} \linebreak
\noindent\hyperlink{RelationToTheEtaFunction}{Relation to the eta-function}\dotfill \pageref*{RelationToTheEtaFunction} \linebreak
\noindent\hyperlink{RelationToAnalyticTorsion}{Relation to analytic torsion}\dotfill \pageref*{RelationToAnalyticTorsion} \linebreak
\noindent\hyperlink{relation_to_prime_geodesic_asymptotics}{Relation to prime geodesic asymptotics}\dotfill \pageref*{relation_to_prime_geodesic_asymptotics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Motivated by the resemblance of the [[Selberg trace formula]] to Weil's formula for the sum of zeros of the [[Riemann zeta function]], (\hyperlink{Selberg56}{Selberg 56}) defined for any compact [[hyperbolic manifold|hyperbolic]] [[Riemann surface]] a [[zeta function]]-like expression, the \emph{[[Selberg zeta function of a Riemann surface]]}. (e.g. \hyperlink{Bump}{Bump, below theorem 19}). There is also a Selberg zeta function ``of odd type'' for odd-dimensional manifolds (\hyperlink{Millson78}{Millson 78}, \hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a, prop. 4.5}).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{for_evendimensional_manifolds}{}\subsubsection*{{For even-dimensional manifolds}}\label{for_evendimensional_manifolds}

(\ldots{})

\hypertarget{DefinitionForOdd}{}\subsubsection*{{For odd-dimensional manifolds}}\label{DefinitionForOdd}

Let

\begin{itemize}%
\item $n \in \mathbb{N}$, $n \geq 1$;


\item $G = SO(n,1)$ the [[Lorentz group]] or $G = Spin(n,1)$ its [[spin group]];


\item $K = SO(n)$ the [[special orthogonal group]] or $K = Spin(n)$ the [[spin group]];


\item $K \hookrightarrow G$ the canonical inclusion, exhibiting a [[maximal compact subgroup]];


\item $G = K A N$ an [[Iwasawa decomposition]];


\item $\Gamma \hookrightarrow G$ a [[discrete group|discrete]] [[subgroup]];


\item $E$ a [[complex vector space|complex]] [[finite dimensional vector space]]


\item $\chi \colon \Gamma \to U(E)$ a [[unitary representation]] of $\Gamma$.



\end{itemize}
Then the [[quotient]]

\begin{displaymath}
X \coloneqq \Gamma \backslash G / K
\end{displaymath}
is a [[hyperbolic manifold]] of odd [[dimension]] with [[fundamental group]] being $\pi_1(X) \simeq \Gamma$. Accordingly the representation $\chi$ is equivalently a [[flat vector bundle]] on $X$.

Write $Conj(\Gamma)$ for the set of [[conjugacy classes]] of $\Gamma$ and write

\begin{displaymath}
Prim(\Gamma) \hookrightarrow Conj(\Gamma)
\end{displaymath}
for the [[subset]] of elements $[g]$ for which $n_\Gamma(g) = 1$. Regarded as elements of the [[fundamental group]] as above, these elements correspond to paths which are [[prime geodesics]] in $X$.

\textbf{Definition}

The \emph{Selberg zeta function} $\zeta_\chi$ of this data is defined for $Re(s)\gt \rho \coloneqq (n-1)/2$ to be the [[infinite product]]

\begin{displaymath}
\zeta_\chi(s) =
  \underset{{[g] \in Prim(\Gamma)} \atop {[g] \neq 1}}{\prod}
  \;
  \underoverset{k = 0}{\infty}{\prod} 
  \det\left(
    1 - 
    e^{-(\rho + s)l(g)}
    S^k(Ad(g)^{-1}_{\mathbf{n}})
    \sigma(m)
    \;
    \chi(g)
  \right)
\end{displaymath}
(\ldots{}) (\hyperlink{BunkeOlbrich94a}{BunkeOlbrich 94a, def. 4.1})

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

\hypertarget{AnalogyWithArtinLFunction}{}\subsubsection*{{Analogy with Artin L-function}}\label{AnalogyWithArtinLFunction}

That the Selberg zeta function is equivalently an [[Euler product]] of [[characteristic polynomials]] is due to (\hyperlink{Gangolli77}{Gangolli 77, (2.72)}, \hyperlink{Fried86}{Fried 86, prop. 5}).

That it is in particular the Euler product of [[characteristic polynomials]] of the [[determinants]] of the [[monodromies]] of the [[flat connection]] corresponding to the given [[group representation]] (similar to the [[Ruelle zeta function]]) is (\hyperlink{BunkeOlbrich94}{Bunke-Olbrich 94, prop. 6.3}) for the even-dimensional case and (\hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a, def. 4.1}) for the odd-dimensional case. (Or rather, the [[Ruelle zeta function]] (\hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a, def. 5.1})).

This is [[analogy|analogous]] to the standard definition of an [[Artin L-function]] if one interprets a) a [[Frobenius map]] $Frob_p$ (as discussed there) as an element of the [[arithmetic fundamental group]] of an [[arithmetic curve]] and b) a [[Galois representation]] as a [[flat connection]].

So under this analogy the Selberg zeta function for hyperbolic 3-manifolds as well as the [[Artin L-function]] for a [[number field]] both are like an [[infinite product]] over primes ([[prime geodesics]] in one case, [[prime ideals]] in the other, see also at \emph{\href{Spec%28Z%29#As3dSpaceContainingKnots}{Spec(Z) -- As a 3-dimensional space containing knots}}) of determinants of monodromies of the given flat connection.

See at \emph{\href{Artin%20L-function#AnalogyWithSelbergZeta}{Artin L-function -- Analogy with Selberg zeta function}} for more. This analogy has been highlighted in (\hyperlink{Brown09}{Brown 09}, \hyperlink{Morishita12}{Morishita 12, remark 12.7}).

\hypertarget{RelationToTheEtaFunction}{}\subsubsection*{{Relation to the eta-function}}\label{RelationToTheEtaFunction}

Under suitable conditions, the Selberg zeta function of odd type is an exponential of the [[eta function]] of a suitable [[Dirac operator]]

\begin{displaymath}
\zeta_S(0) = \exp\left(i \pi \eta_D(0)\right)
\end{displaymath}
(\hyperlink{Millson78}{Millson 78}, \hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a, prop. 4.5}, \hyperlink{Park01}{Park 01, theorem 1.2}, \hyperlink{GuillarmouMoroianuPark09}{Guillarmou-Moroianu-Park 09}).

\hypertarget{RelationToAnalyticTorsion}{}\subsubsection*{{Relation to analytic torsion}}\label{RelationToAnalyticTorsion}

The [[Ruelle zeta function]] at 0 gives a power of [[analytic torsion]]

(\hyperlink{Fried86}{Fried 86}, \hyperlink{BunkeOlbrich94a}{Bunke-Olbrich 94a, theorem 5.5.})

\begin{itemize}%
\item \href{analytic+torsion#RelationToSelbergZeta}{analytic torsion -- relation to Selberg zeta function}

\end{itemize}
\hypertarget{relation_to_prime_geodesic_asymptotics}{}\subsubsection*{{Relation to prime geodesic asymptotics}}\label{relation_to_prime_geodesic_asymptotics}

The Selberg zeta function controls the [[asymptotics]] of [[prime geodesics]] via the [[prime geodesic theorem]] in direct [[analogy]] to how the [[Riemann zeta function]] controls the asymptotics of [[prime numbers]] via the [[prime number theorem]].

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

\begin{itemize}%
\item [[Ruelle zeta function]]

\end{itemize}
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

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

The original article is

\begin{itemize}%
\item [[Atle Selberg]], \emph{Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series}, Journal of the Indian Mathematical Society 20 (1956) 47-87.

\end{itemize}
Review includes

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Selberg_zeta_function}{Selberg zeta function}}


\item [[Matthew Watkins]], citation collection on \emph{\href{http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics4.htm}{Selberg trace formula and zeta functions}}


\item Bump, below theorem 19 in \emph{Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$} ([[BumpSpectralTheory.pdf:file]])


\item \emph{Selberg and Ruelle zeta functions for compact hyperbolic manifolds} (\href{http://www.math.uni-bonn.de/people/xenia/Oberseminar%202014/Oberseminar%202014.pdf}{pdf})



\end{itemize}
Expression of the Selberg/Ruelle zeta function as an [[Euler product]] of [[characteristic polynomials]] is due to

\begin{itemize}%
\item Ramesh Gangolli, \emph{Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one}, Illinois J. Math. Volume 21, Issue 1 (1977), 1-41. (\href{http://projecteuclid.org/euclid.ijm/1256049498}{Euclid})


\item [[David Fried]], \emph{The zeta functions of Ruelle and Selberg. I}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure, S\'e{}r. 4, 19 no. 4 (1986), p. 491-517 (\href{http://www.numdam.org/item?id=ASENS_1986_4_19_4_491_0}{Numdam})



\end{itemize}
The analogy with the [[Artin L-function]] is highlighted in

\begin{itemize}%
\item Darin Brown, \emph{Lifting properties of prime geodesics}, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 (\href{http://projecteuclid.org/euclid.rmjm/1239113439}{euclid})


\item [[Masanori Morishita]], section 12.1 of \emph{Knots and Primes: An Introduction to Arithmetic Topology}, 2012 (\href{https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false}{web})



\end{itemize}
Discussion of the relation between, on the one hand, [[zeta function of an elliptic differential operator|zeta function]] of [[Laplace operators]]/[[eta function of a self-adjoint operator|eta funcstions]] of [[Dirac operators]] and, on the other hand, Selberg zeta functions includes

\begin{itemize}%
\item [[Eric D'Hoker]] [[Duong Phong]], \emph{Communications in Mathematical Physics}, Volume 104, Number 4 (1986), 537-545 (\href{http://projecteuclid.org/euclid.cmp/1104115166}{Euclid})


\item [[Peter Sarnak]], \emph{Determinants of Laplacians}, Communications in Mathematical Physics, Volume 110, Number 1 (1987), 113-120. (\href{http://projecteuclid.org/euclid.cmp/1104159171}{Euclid})


\item [[Ulrich Bunke]], [[Martin Olbrich]], Andreas Juhl, \emph{The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function}, Annals of Global Analysis and Geometry February 1994, Volume 12, Issue 1, pp 357-405


\item [[Ulrich Bunke]], Martin Olbrich, \emph{Theta and zeta functions for locally symmetric spaces of rank one} (\href{http://arxiv.org/abs/dg-ga/9407013}{arXiv:dg-ga/9407013})



\end{itemize}
and for odd-dimensional spaces also in

\begin{itemize}%
\item [[David Fried]], \emph{Analytic torsion and closed geodesics on hyperbolic manifolds}, Invent. math. 84, 523-540 (1986) (\href{http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN356556735_0084&divID=LOG_0026&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN356556735_0084%7C&targetFileName=PPN356556735_0084_LOG_0026.pdf&}{pdf})


\item [[John Millson]], \emph{Closed geodesic and the $\eta$-invariant}, Ann. of Math., 108, (1978) 1-39 (\href{http://www.jstor.org/stable/1970928}{jstor})


\item [[Ulrich Bunke]], [[Martin Olbrich]], \emph{Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one} (\href{http://arxiv.org/abs/dg-ga/9407012}{arXiv:dg-ga/9407012})


\item [[Ulrich Bunke]], [[Martin Olbrich]] \emph{$\Gamma$-Cohomology and the Selbeg zeta function} (\href{http://arxiv.org/abs/dg-ga/9411004}{arXiv:dg-ga/9411004})


\item [[Ulrich Bunke]], [[Martin Olbrich]], \emph{Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group} (\href{http://arxiv.org/abs/dg-ga/9603003}{arXiv:dg-ga/9603003})


\item [[Ulrich Bunke]], [[Martin Olbrich]], \emph{Selberg zeta and theta functions: a differential operator approach}, Akademie Verlag 1995


\item Jinsung Park, \emph{Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps} (\href{http://arxiv.org/abs/math/0111175}{arXiv:0111175})


\item [[Joshua Friedman]], \emph{The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations} (\href{http://arxiv.org/abs/math/0410067}{arXiv:math/0410067})


\item [[Joshua Friedman]], \emph{Regularized determinants of the Laplacian for cofinite Kleinian groups with finite-dimensional unitary representations}, Communications in Mathematical Physics (\href{http://arxiv.org/abs/math/0605288}{arXiv:math/0605288})


\item Colin Guillarmou, Sergiu Moroianu, Jinsung Park, \emph{Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds} (\href{http://arxiv.org/abs/0901.4082}{arXiv:0901.4082})



\end{itemize}
[[!redirects Selberg zeta function of a Riemann surface]]



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