PROFINITE GROUPS WITH MULTIPLICATIVE PROBABILISTIC ZETA FUNCTION

Abstract

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)\,{=}\,\sum_{n}{a_n}/{n^s}$ is associated, where $a_n \,{=}\,\sum_{|G:H|=n} \mu_G(H)$. It is proved that $G$ is prosoluble if and only if the sequence $\{a_n\}_{n \in \mathbb N}$ is multiplicative, that is, $a_{rs}\,{=}\,a_ra_s$ for any pair of coprime positive integers $r$ and $s$. This extends the analogous result on the probabilistic zeta function of finite groups.