Normal zeta function of a group

Definition
Let $$G$$ be a group. The normal 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 normal subgroups of $$G$$ of index $$n$$. Equivalently, it is:

$$\sum_{H \underline{\triangleleft}_f G} [G:H]^{-s}$$

summing up over all normal 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 (see also group with finitely many homomorphisms to any finite group).

Related notions

 * Zeta function of a group