On Hasse zeta functions of group algebras of almost nilpotent groups (Q1300302)

The purpose of this paper is to give a proof of a theorem concerning the convergence of Hasse's zeta-functions of group algebra of almost nilpotent groups (= a group which has a nilpotent subring of finite index), which is a generalization of Hasse's zeta-functions of commutative rings by the author. The main theorem is the following: Let \(G\) be a finitely generated group which has a nilpotent subring of finite index, \(R\) a finitely generated commutative ring over \(\mathbb{Z}\) and \(A\) the group ring \(R[G]\). Then Hasse's zeta-function \(\zeta_A(s)\) of \(A\) is absolutely convergent for \(s\in \mathbb{C}\) with sufficiently large real part.