Zeta function of a group

Definition

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

$\zeta _{G}(s)=\sum _{n=1}^{\infty }a_{n}(G)n^{-s}$

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

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

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.