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 <email@example.com>