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PROFINITE GROUPS WITH MULTIPLICATIVE PROBABILISTIC ZETA FUNCTION
E. DETOMI, A. LUCCHINI
Journal:

Journal of the London Mathematical Society / Volume 70 / Issue 1 / August 2004

Published online by Cambridge University Press:

23 July 2004, pp. 165-181

Print publication:

August 2004
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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.