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 sₙα(G) = sumₙ₌₁ⁿ aₙ(G)/nα. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence α(G) of ζG(s) is a rational number and ζG(s) can be meromorphically continued to Re(s)>α(G)-δfor some δ>0. The continued function is holomorphic on the line (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 sₙ(G) c Nα⁽ᵍ⁾(N)ᵇ⁽ᵍ⁾ sₙα⁽ᵍ⁾(G) c' (N)ᵇ⁽ᵍ⁾⁺¹ for N→ ∞.