Zeta function of a Lie ring

Definition
Let $$L$$ be a Lie ring. The zeta function of $$L$$ is defined as:

$$\zeta_L(s) = \sum_{n=1}^{\infty} a_n(L)n^{-s}$$

where $$a_n(L)$$ denotes the number of Lie subrings of $$L$$ of index $$n$$. Equivalently, it is:

$$\sum_{S \le_f L} [L:S]^{-s}$$

summing up over all Lie subrings of finite index in $$L$$.

The coefficients $$a_n(L)$$ are all finite when the Lie ring $$L$$ is finitely generated.

Related notions

 * Zeta function of a group
 * Normal zeta function of a group
 * Ideal zeta function of a Lie ring