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:

$\sum_{H \le_f G} [G:H]^{-s}$

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

The coefficients $a_n(G)$ are all finite when the group $G$ is finitely generated. This follows from finitely generated implies finitely many homomorphisms to any finite group and equivalence of definitions of group with finitely many homomorphisms to any finite group (in turn a consequence of Poincare's theorem).

Facts

Convergence

When the group is a PSG-group (i.e., 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.