Let $G$ be a group with a finite symmetric set $S$ of generators. Let $\ell_S(x)$ denote the word-length of a given $x\in G$. For $s\in\mathbb C$ set $$ Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s}, $$ where $G^*=G\smallsetminus\{1\}$. It may happen that this sum converges for some $s$. It does not converge for free groups but does converge for abelian groups. In the few simple cases I have computed, it turned out to be a linear combination of Riemann zeta functions with shifted arguments.

My question is a reference request: Has this kind of Dirichlet series been investigated? If so, I would like to have references.

Thank you.

Let $G$ be a group with a finite symmetric set $S$ of generators. Let $\ell_S(x)$ denote the word-length of a given $x\in G$. For $s\in\mathbb C$ set $$ Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s}, $$ where $G^*=G\smallsetminus\{1\}$. It may happen that this sum converges for some $s$. By Gromov's Theorem on groups of polynomial growth, this is the case if and only if the group has a nilpotent subgroup of finite index. In the very few cases I have computed, this function turned out to be a linear combination of Riemann zetas with shifted arguments.

My question is a reference request: Has this kind of Dirichlet series been investigated? If so, I would like to have references.

Thank you.