Let $G$ be a finitely generated group and let $\mathcal{X}$ be a family of finite index subgroups of $G$. Define

Note that these numbers are finite since a finitely generated group has only finitely many subgroups of a given index. We define the zeta function of the family $\mathcal{X}$ to be the formal Dirichlet series

Two important special cases are the zeta function counting all subgroups and the zeta function counting normal subgroups. Let $\mathcal{S}(G)$ and $\mathcal{N}(G)$ be the families of all finite index subgroups of $G$ and of all finite index normal subgroups of $G$, respectively. We write $a_{n}^{\leqslant}(G)=a_{n}(\mathcal{S}(G))$ and $a_{n}^{\trianglelefteqslant}(G)=a_{n}(\mathcal{N}(G))$ and define

and

If, in addition, $G$ is nilpotent, then $\zeta_{G}^{\leqslant}$ has a decomposition as a formal Euler product

where

An analogous result holds for the normal zeta function $\zeta_{G}^{\trianglelefteqslant}$. The result for both $\zeta_{G}^{\leqslant}$ and $\zeta_{G}^{\trianglelefteqslant}$ can be proved using properties of the profinite completion of $G$. However, a simpler proof for the normal zeta function is provided by the fact that a finite nilpotent group decomposes into a direct product of its Sylow subgroups. These results allow the zeta functions to be expressed in terms of $p$-adic integrals, which can in turn be used to prove (using some high-powered machinery) that $\zeta_{{G,p}}^{\leqslant}(s)$ and $\zeta_{{G,p}}^{\trianglelefteqslant}(s)$ are rational functions in $p$ and $p^{{-s}}$.

In the case when $G$ is a $\mathcal{T}$-group, that is, $G$ is finitely generated, torsion free, and nilpotent, define $\alpha_{G}^{\leqslant}$ to be the abscissa of convergence of $\zeta_{G}^{\leqslant}$. That is, $\alpha_{G}^{\leqslant}$ is the smallest $\alpha\in\mathbb{R}$ such that $\zeta_{G}^{\leqslant}$ defines a holomorphic function in the right half-plane $\{z\in\mathbb{C}\mid\Re(z)>\alpha\}$. It can then be shown that $\alpha_{G}^{\leqslant}\leqslant\mathrm{h}(G)$, where $\mathrm{h}(G)$ is the Hirsch number of $G$. Therefore, if $G$ is a $\mathcal{T}$-group, $\zeta_{G}^{\leqslant}$ defines a holomorphic function in some right half-plane.