Zeta function of a group

Definition

Let $G$ be a group. The zeta function of $G$ is defined as:

${\zeta }_G(s)=\sum _{n=1}^{\mathrm{\infty }}{a}_n(G)n^{-s}$

where $a_n(G)$ denotes the number of subgroups of $G$ of index $n$. Equivalently, it is:

${\sum }_{H{\le }_{f}G}[G:H{]}^{-s}$

summing up over all subgroups of finite index in $G$.

Facts

Convergence

When the group is a PSG-group (viz, it has polynomial subgroup growth) then the zeta function is well-defined and is convergent for a value of $s$ for which $n^{Re(s)}/a_n(G)$ grows at a rate that is strictly more than linear (for instance, quadratic or more). The zeta function is not convergent for other values of $s$.

Euler product

When the group is a finitely generated torsion-free nilpotent group, we can get an Euler product formula, for the zeta function:

${\zeta }_G(s)=\prod {\zeta }_G,p(s)$

where ${\zeta }_{G,p}(s)={\sum }_n{a}_{p^n}(G)p^{-ns}$

This is a consequence of the fact that any finite nilpotent group is a direct product of its Sylow subgroups.