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Analytic properties of zeta functions and subgroup growth | | Marcus du Sautoy
; Fritz Grunewald
; | | Date: |
1 Nov 2000 | | Journal: | Ann. of Math. (2) 152 (2000), no. 3, 793--833 | | Subject: | Group Theory | math.GR | | Abstract: | 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 alpha a real number and N a nonnegative integer, define s_N^alpha(G) = sum_{n=1}^N a_n(G)/n^alpha. Main Theorem: Let G be a finitely generated nilpotent infinite group. (1) The abscissa of convergence alpha(G) of zeta_G(s) is a rational number and zeta_G(s) can be meromorphically continued to Re(s)>alpha(G)-delta for some delta >0. The continued function is holomorphic on the line Re(s) = (alpha)G except for a pole at s=alpha(G). (2) There exist a nonnegative integer b(G) and some real numbers c,c’ such that s_{N}(G) ~ c N^{alpha(G)}(log N)^{b(G)} s_{N}^{alpha(G)}(G) ~ c’ (log N)^{b(G)+1} for N
ightarrow infty . | | Source: | arXiv, math.GR/0011267 | | Services: | Forum | Review | PDF | Favorites | |
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