Zeta function of a group

Let $G$ be a finitely generated group and let $𝒳$ be afamily of finite index subgroups of $G$. Define

$$a_n(𝒳)=|\{H\in 𝒳\mid |G:H|=n\}|.$$

Note that these numbers are finite since a finitely generated grouphas only finitely many subgroups of a given index. We define thezeta function of the family $𝒳$ to be the formalDirichlet series

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

Two important special cases are the zeta function counting allsubgroups and the zeta function counting normal subgroups. Let$𝒮(G)$ and $𝒩(G)$ be the families of all finiteindex subgroups of $G$ and of all finite index normal subgroups of$G$, respectively. We write $a_n^{⩽}(G)=a_n(𝒮(G))$ and$a_n^{⊴}(G)=a_n(𝒩(G))$ and define

$${\zeta }_G^{⩽}(s)={\zeta }_{𝒮(G)}(s)=\sum _{H{⩽}_{\mathrm{f}}G}|G:H{|}^{-s},$$

and

$${\zeta }_G^{⊴}(s)={\zeta }_{𝒩(G)}(s)=\sum _{N{⊴}_{\mathrm{f}}G}|G:N{|}^{-s}.$$

If, in addition, $G$ is nilpotent, then ${\zeta }_G^{⩽}$ has adecomposition as a formal Euler product

$${\zeta }_G^{⩽}(s)=\prod _{p\text{ prime}}{\zeta }_{G,p}^{⩽}(s),$$

where

$${\zeta }_{G,p}^{⩽}(s)=\sum _{i=0}^{\mathrm{\infty }}{a}_{p^i}^{⩽}(G)p^{-is}.$$

An analogous result holds for the normal zeta function${\zeta }_G^{⊴}$. The result for both ${\zeta }_G^{⩽}$ and${\zeta }_G^{⊴}$ can be proved using properties of the profinitecompletion of $G$. However, a simpler proof for the normal zetafunction is provided by the fact that a finite nilpotent groupdecomposes into a direct product of its Sylow subgroups. These resultsallow the zeta functions to be expressed in terms of $p$-adicintegrals, which can in turn be used to prove (using some high-poweredmachinery) that ${\zeta }_{G,p}^{⩽}(s)$ and ${\zeta }_{G,p}^{⊴}(s)$ arerational functions in $p$ and $p^{-s}$.

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