1. Introduction

Let G be a group, and write an(G) for the number of subgroups of index n in G. If G is a profinite group, we count only open subgroups. When G is finitely generated (either abstractly or topologically), an(G) is finite for all n∈N. In this case we can encode the arithmetic function n[map ]an(G) in the following zeta function:

formula here

This function was introduced by Grunewald, Segal and Smith [9] and subsequently studied in [1–6, 7, 11].