Profinite groups in which the probabilistic zeta function coincides with the subgroup zeta function

The Probabilistic Zeta Function

This paper is a summary of results on the PG(s) function, which is the reciprocal of the probabilistic zeta function for finite groups. This function gives the probability that s randomly chosen elements generate a group G, and information about the structure of the group G is embedded in it.

The Fibonacci Zeta Function

We consider the lacunary Dirichlet series obtained by taking the reciprocals of the s-th powers of the Fibonacci numbers. This series admits an analytic continuation to the entire complex plane. Its special values at integral arguments are then studied. If the argument is a negative integer, the value is algebraic. If the argument is a positive even integer, the value is transcendental by Neste...

a finiteness condition on the coefficients of the probabilistic zeta function

we discuss whether finiteness properties of a profinite group $g$ can be deduced from the coefficients of the probabilisticzeta function $p_g(s)$. in particular we prove that if $p_g(s)$ is rational and all but finitely many non abelian composition factors of $g$ are isomorphic to $psl(2,p)$ for some prime $p$, then $g$ contains only finitely many maximal subgroups.

Mollifying the Riemann Zeta-function

The Divisor of Selberg's Zeta Function for Kleinian Groups

We compute the divisor of Selberg's zeta function for convex co-compact, torsion-free discrete groups acting on a real hyperbolic space of dimension n + 1. The divisor is determined by the eigenvalues and scattering poles of the Laplacian on X = nH together with the Euler characteristic of X compacti ed to a manifold with boundary. If n is even, the singularities of the zeta funciton associated...