Profinite groups in which the probabilistic zeta function has no negative coefficients

To a finitely generated profinite group G, a formal Dirichlet series PG(s) =σn ϵNan(G)/ns is associated, where an(G) =σ|G:H|=nμ(H,G) and μ(H,G) denotes the Möbius function of the lattice of open subgroups of G. Its formal inverse (PG(s))-1 is the probabilistic zeta function of G. When G is prosoluble, every coefficient of (PG(s))-1 is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.