Profinite groups with nonabelian crowns of bounded rank and their probabilistic zeta function

It has been conjectured by Mann that the infinite sum ∑ μ(H,G)/|G:H|^ s , where H ranges over all open subgroups of a finitely generated profinite group G, converges absolutely in some half right plane if G is positively finitely generated. We prove that the conjecture is true if the nonabelian crowns of G have bounded rank. In particular Mann’s conjecture holds if G has polynomial subgroup growth or is an adelic profinite group.