Zeta function of the Burnside ring for cyclic groups (Q2911125)

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 \(n\geq 3\).