Bounds for special values of Selberg zeta functions of Riemann surfaces
type: Item
instance of: scholarly article
title: Bounds for special values of Selberg zeta functions of Riemann surfaces
zbMATH Open document ID: 0986.11058
author: Jürg Kramer
author: Jay Jorgenson
publication date: 2001-12-03T00:00:00Z
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 1678295
zbMATH Keywords: constant term of logarithmic derivative
zbMATH Keywords: Selberg zeta function
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: lower bound
zbMATH Keywords: upper bounds
zbMATH Keywords: length spectrum
zbMATH Keywords: small eigenvalues
zbMATH Keywords: hyperbolic Laplacian
zbMATH Keywords: congruence subgroups
MaRDI profile type: Publication
full work available at URL: crll.2001.092
OpenAlex ID: W2038960689
cites work: Geometry and spectra of compact Riemann surfaces
cites work: Q3856819
cites work: Q3975268
cites work: Basic analysis of regularized series and products
cites work: Q4321409
cites work: Q5625265
DOI: 10.1515/CRLL.2001.092
published in: Journal für die Reine und Angewandte Mathematik
review text: The authors give lower and upper bounds for the constant term \(c_M\) of the logarithmic derivative of the Selberg zeta function \(Z_M(s)\) at \(s=1\) for Riemann surfaces \(M= \Gamma\setminus\mathbb{H}\), where \(\Gamma \subseteq \text{PSL}_2(\mathbb{R})\) is any Fuchsian group of the first kind. The lower bound for \(c_M\) is essentially of the form \(O(\log \operatorname {vol}(M))\) with \(\operatorname {vol}(M)\) denoting the hyperbolic volume of \(M\). The upper bounds are of a more complicated nature; they involve the length spectrum of \(M\) and the small eigenvalues of the hyperbolic Laplacian. If \(M\) is a finite cover of a fixed base space \(M_0\) and \(\Gamma\) is an arithmetic subgroup, the upper bound for \(c_M\) can be simplified to \(O_{M_0} (\operatorname {vol}(M)^{1+\varepsilon})\) for any \(\varepsilon>0\). In particular, for the congruence subgroups \(\Gamma_0(N)\) with square-free \(N\), the upper bound can be improved to \(O(N^\varepsilon)\).
description: scientific article; zbMATH DE number 1678295