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

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)={\sum }_{n\in \mathbb{N}}{a}_n(G)\mathrm{/}{n}^s$ is associated, where $a_n(G)={\sum }_{\mathrm{\mid }G:H\mathrm{\mid }=n}\mu (H,G)$ and $\mu (H,G)$ denotes the M"obius function of the lattice of open subgroups of $G\mathrm{.}$ Its formal inverse $P_G^{-1}(s)$ is the probabilistic zeta function of $G$. When $G$ is prosoluble, every coefficient of $(P_G(s){)}^{-1}$ is nonnegative. In this paper we discuss the general case and we produce % existence of a non-prosoluble example and We construct a non-prosoluble finitely generated group $G$ with the same property.