Arithmetic forms of Selberg zeta functions with applications to prime geodesic theorem
type: Item
instance of: scholarly article
title: Arithmetic forms of Selberg zeta functions with applications to prime geodesic theorem
zbMATH Open document ID: 1027.11065
author: T. Arakawa
author: Shin-ya Koyama
author: Maki Nakasuji
published in: Proceedings of the Japan Academy. Series A
publication date: 2004-01-14T00:00:00Z
review text: Excerpts from portions of the introduction: ``Let \(\Gamma\) be a discrete subgroup of \(\text{SL}_2(\mathbb{R})\) containing \(-1_2\) with finite covolume \(v(\Gamma\setminus{\mathfrak H})\), \({\mathfrak H}\) denoting the upper half plane. The Selberg zeta-function attached to \(\Gamma\) is defined by \[ Z_\Gamma(s):= \prod_{\{P\}_\Gamma} \prod_{m=0}^\infty (1-N(P)^{-s-m}), \quad(\operatorname {Re}(s)>1), \] where \(\{P\}_\Gamma\) runs through all primitive hyperbolic conjugacy classes of \(\Gamma\) with \(\text{tr}(P)>2\), and \(N(P):= |\rho|^2\) with \(\rho\) the eigenvalue of \(P\in \Gamma\) such that\(|\rho|> 1\). \dots{} For any basis \(\{u_i\}\) of \({\mathcal O}\) [a maximal order of \(B\), an indefinite division quaternion algebra over \(\mathbb{Q}\)] over \(\mathbb{Z}\), set \[ d(B)= |\det (\operatorname {tr}(u_iu_j))|^{1/2}. \] Put \[ {\mathcal D}:= \{D\in \mathbb{Z}_{>0}\mid D\equiv 0,1\pmod 4,\text{ not a square}\}. \] Let \({\mathfrak o}\) be an order of \(K= \mathbb{Q}(\sqrt{D})\) and \(h({\mathfrak o})= h(D)\) be the number of classes of proper \({\mathfrak o}\)-ideals in the narrow sense. We moreover set \[ \lambda(K)= \prod_{p\mid d(B)} \Bigl(1 -\bigl(\tfrac Kp\bigr)\Bigr), \] where \((K/p)\) denotes the Artin symbol for \(K= \mathbb{Q}(\sqrt{D})\). Let \(\varepsilon_D= (\alpha+ \beta\sqrt{D})/2\) with \((\alpha,\beta)\) being the minimal solution of the Pell equation: \(x^2- Dy^2=4\). The main theorem of this paper is as follows. Theorem 1.1. Let \(B\) be a division indefinite quaternion algebra over \(\mathbb{Q}\). Then \[ Z_B(s)= \mathop{{\prod}^*}_{D>0} \prod_{n=0}^\infty \Bigl(1-\varepsilon_D^{-2(s+n)} \Bigr)^{h(D)\lambda(D)}, \] and \[ \frac{Z_B'}{Z_B}(s)= \mathop{{\sum}^*}_{D>0} \sum_{m=1}^\infty h(D) \lambda(D) \log \varepsilon_D^2\cdot \frac{\varepsilon_D^{-2ms}} {1-\varepsilon_D^{-2m}}, \] where \(\lambda(D)= \lambda(\mathbb{Q} (\sqrt{D}))\) and the symbol * indicates that \(D\) runs through all elements in \({\mathcal D}\) satisfying th…
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 58E10
zbMATH DE Number: 1902571
zbMATH Keywords: Selberg zeta-function
zbMATH Keywords: prime geodesic theorem
zbMATH Keywords: number of primitive hyperbolic conjugacy classes
zbMATH Keywords: division indefinite quaternion algebra
reviewed by: Mark Sheingorn
MaRDI profile type: Publication
full work available at URL: pjaa.78.120
OpenAlex ID: W2029135115
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DOI: 10.3792/PJAA.78.120
description: scientific article; zbMATH DE number 1902571