Spectral functions, special functions and the Selberg zeta function
type: Item
instance of: scholarly article
title: Spectral functions, special functions and the Selberg zeta function
zbMATH Open document ID: 0631.10025
DOI: 10.1007/BF01212422
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: The author starts from an arbitrary sequence \((\lambda_k)_{k\geq 1}\) of positive real numbers such that \(\lambda_k\to \infty\) which is subject to suitable regularity conditions. (Typically, \((\lambda_k)_{k\geq 1}\) will be the spectrum of a differential operator or a number-theoretically defined sequence.) Then he forms the associated \(\theta\)-series, the Fredholm determinant, the zeta-function and the functional determinant and he discusses the relations between these functions by means of regularization techniques. In particular, he establishes a very general relation between the functional determinant and the Fredholm determinant. The results apply to a wide variety of examples covering many classical situations from mathematics and theoretical physics. In particular, the spectral sequence of the Laplacian on the two-dimensional sphere is related with the Barnes \(G\)-function. The main application is an explicit factorization of the Selberg zeta-function into two functional determinants, one of which is expressible in terms of the Barnes \(G\)-function. Relations with some recent results on the Selberg zeta-function are also established.
reviewed by: Jürgen Elstrodt
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 35P99
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J52
Mathematics Subject Classification ID: 11F72
zbMATH DE Number: 4025522
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: compact Riemann surface
zbMATH Keywords: Glaisher-Kinkelin constant
zbMATH Keywords: Fredholm determinant
zbMATH Keywords: zeta-function
zbMATH Keywords: functional determinant
zbMATH Keywords: spectral sequence
zbMATH Keywords: Laplacian
zbMATH Keywords: Barnes G-function
zbMATH Keywords: factorization of the Selberg zeta function
author: André Voros
MaRDI profile type: Publication
cites work: On determinants of Laplacians on Riemann surfaces
cites work: Analytic torsion for complex manifolds
cites work: On the Analytic Torsion and Eta Invariant for Negatively Curved Manifolds
cites work: Analytic torsion and closed geodesics on hyperbolic manifolds
cites work: What is the Laplace Transform?
cites work: The spectrum of positive elliptic operators and periodic bicharacteristics
cites work: Q3041568
cites work: Q3285751
cites work: The Selberg trace formula and the Riemann zeta-function
cites work: Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen
cites work: Toeplitz Determinants with Singular Generating Functions
cites work: Q5788558
cites work: Q3671491
cites work: Determinants of Laplacians
cites work: Spectral fluctuations and zeta functions
full work available at URL: https://doi.org/10.1007/bf01212422
OpenAlex ID: W2044405326
description: scientific article; zbMATH DE number 4025522