Analytic properties of Euler products of Igusa-type zeta functions and subgroup growth of nilpotent groups

We prove asymptotic results for the number of subgroups of index at most n is an element of N in a finitely generated nilpotent group Gamma. These are obtained by applying a Tauberian theorem to the zeta-function of Gamma. To be able to do this we show that this Dirichlet series and also certain Eul er products which arise from local Igusa-type p-adic integrals can be merom orphically continued past their abscissa of convergence. (C) 1999 Academie des Sciences/Editions scientifiques et medicales Elsevier SAS.