Profinite groups with multiplicative probabilistic zeta function

Given a finitely generated profinite group G, a formal Dirichlet series P_G(s) is associated to G with the property that, if the series converges, then its value in large positive integers s is the probability that G is generated by s random elements. The formal inverse of this series is called the `probabilistic zeta function' of G. The present paper shows that G is prosoluble if and only if P_G(s) is multiplicative. This extends the analogous result on the probabilistic zeta function of finite groups.