Zeta function of the Burnside ring for cyclic groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 1258.19001
author: David Villa-Hernandez
publication date: 2012-09-12T00:00:00Z
full work available at URL: index.html
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 6081567
zbMATH Keywords: Burnside rings
zbMATH Keywords: zeta functions
zbMATH Keywords: fiber product
MaRDI profile type: Publication
title: Zeta function of the Burnside ring for cyclic groups
published in: International Journal of Algebra
review text: Let \(p\) be a prime number, \(C_{p^n}\) the cyclic groups of order \(p^n\) and \(B(C_{p^n})\) the Burnside ring of \(C_{p^n}\). In this paper, using the method established by Bushnell and Reiner, the author explicitly calculates the zeta functions \(\zeta_{\Gamma_{n}}(s)\) for \(n=1,2\) where \(\Gamma_{n}:=\mathbb{Z}_{\{p\}} \otimes_{\mathbb{Z}}B(C_{p^n})\) with \(\mathbb{Z}_{\{p\}}\) being the localization of \(\mathbb{Z}\) at \(p\). The results are given as follows: NEWLINE\[NEWLINE \begin{aligned} \zeta_{\Gamma_{1}}(s) &=\frac{1-p^{-s}+p^{1-2s}}{(1-p^{-s})^{2}},\\ \zeta_{\Gamma_{2}}(s) &=\frac{1-2p^{-s}+(1+p+p^2)p^{-2s}-2p^{1-3s}+(1-p+p^2)p^{1-4s}+(-1+p)p^{2-5s}}{(1-p^{-s})^{3}}. \end{aligned} NEWLINE\]NEWLINE Note that these have been already obtained in the author's previous paper [Commun. Algebra 37, No. 5, 1758--1786 (2009; Zbl 1241.11110)], where the expressions are established by systematically finding all ideals of finite index of the \(\Gamma_{n}\). The author also notices that this method enables us to recursively compute \(\zeta_{\Gamma_{n}}(s)\) for \(n\geq 3\).
reviewed by: Yoshinori Yamasaki
description: scientific article; zbMATH DE number 6081567