Analytic properties of zeta functions and subgroup growth
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In this paper we introduce some new methods to understand the analytic behaviour of the zeta function of a group. We can then combine this knowledge with suitable Tauberian theorems to deduce results about the growth of subgroups in a nilpotent group. In order to state our results we introduce the following notation. For αa real number and N a nonnegative integer, define
s_N^α(G) = sum_{n=1}^N a_n(G)/n^α.
Main Theorem: Let G be a finitely generated nilpotent infinite group.
(1) The abscissa of convergence α(G) of ζ_G(s) is a rational number and ζ_G(s) can be meromorphically continued to Re(s)>α(G)-δfor some δ>0. The continued function is holomorphic on the line \Re(s) = (α)G except for a pole at s=α(G).
(2) There exist a nonnegative integer b(G) and some real numbers c,c' such that
s_{N}(G) ~ c N^{α(G)}(\log N)^{b(G)}
s_{N}^{α(G)}(G) ~ c' (\log N)^{b(G)+1}
for N\rightarrow \infty .
41 pages, published version, abstract added in migration
41 pages, published version, abstract added in migration