On the Epstein zeta function
type: Item
instance of: scholarly article
title: On the Epstein zeta function
zbMATH Open document ID: 0810.11022
DOI: 10.1007/BF03323415
published in: Results in Mathematics
publication date: 1995-04-11T00:00:00Z
review text: An elementary proof is given of the range of convergence of Epstein zeta functions \[ \zeta(T:s):= \sum_{0\neq g\in \mathbb{Z}^ n} (g^ t Tg)^{- s}, \] with \(T\) any real symmetric positive-definite \(n\times n\) matrix. In particular, a very simple proof of the following theorem is presented: the series converges absolutely for \(\text{Re} (s)> n/2\). The first proof of such theorem was due to \textit{G. Eisenstein} [J. Reine Angew. Math. 35, 137-184 (1847)] and some other proofs are also noted in the text. The new demonstration is very simple and is done by induction on \(n\). As a corollary it is seen in the paper that the above function admits a meromorphic continuation to the whole complex \(s\)-plane with a simple pole at \(s= n/2\), which is also a well-known result. Related brand new results on non-homogeneous Epstein zeta functions (in particular, an extension of the famous Chowla-Selberg formula) have been found by \textit{E. Elizalde} [J. Math. Phys. 35, 6100-6122 (1994), and an unpublished preprint].
Mathematics Subject Classification ID: 11E45
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 604858
zbMATH Keywords: analytic continuation
zbMATH Keywords: range of convergence
zbMATH Keywords: Epstein zeta functions
author: Aloys Krieg
reviewed by: Emilio Elizalde
MaRDI profile type: Publication
cites work: Q3134751
cites work: Q4021268
cites work: Q3881045
cites work: On Epstein's Zeta-function.
cites work: Q3693558
cites work: Q3994578
cites work: Q3909097
description: scientific article; zbMATH DE number 604858