On the Probabilistic Zeta Function for Finite Groups

LetGbe a finite group, and define the function[formula]where μ is the Möbius function on the subgroup lattice ofG. The functionP(G, s) is the multiplicative inverse of a zeta function forG, as described by Mann and Boston. Boston conjectured thatP′(G, 1) = 0 ifGis a nonabelian simple. We will prove a generalization of this conjecture, showing thatP′(G, 1) = 0 unlessG/Op(G) is cyclic for some primep.