Analytic properties of zeta functions and subgroup growth
| dc.creator | Sautoy, Marcus du | |
| dc.creator | Grunewald, Fritz | |
| dc.date | 2000-11-01 | |
| dc.date.accessioned | 2026-07-07T04:38:57Z | |
| dc.date.available | 2026-07-07T04:38:57Z | |
| dc.description | 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 . | |
| dc.description | 41 pages, published version, abstract added in migration | |
| dc.identifier | https://arxiv.org/abs/math/0011267 | |
| dc.identifier | http://arxiv.org/abs/math/0011267 | |
| dc.identifier | Ann. of Math. (2) 152 (2000), no. 3, 793--833 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60480 | |
| dc.subject | Group Theory | |
| dc.title | Analytic properties of zeta functions and subgroup growth | |
| dc.type | text |