A probabilistic generalization of the Riemann zeta function, by Nigel Boston

This paper has appeared in Analytic number theory, vol. 1 (Allerton Park, IL, 1995), 155-162, Progr. Math. 138, Birkhauser Boston, Boston, MA, 1996, and so the dvi version has been removed. Let G be a finitely generated profinite group. The probability P(G,s) that G is generated by s elements, (formally at least) a Dirichlet series, is studied, in particular its factorization into generalized Euler factors and the zeros of these factors. One amusing conjecture that appears is that P(G,s) has a double zero at s=1 if G is a finite, non-abelian simple group. Since posting this paper, a generalized version of this conjecture has been proven by John Shareshian. P(A_6,s) even has a triple zero at s=1.

Nigel Boston <boston@math.uiuc.edu>