Zeta functions are being developed for all sorts of mathematical objects these days. One general situation is that of zeta functions of groups. If $G$ is a finitely generated group then we let
$$\zeta_G(s):=\sum_{H\le_{\Large f} G}[G:H]^{-s}=\sum_{n\ge1}a_n(G)\,n^{-s}\tag{1}$$ $$\text{where}\quad a_n(G):=\#\{H\le G: [G:H]=n\}. \tag{2}$$
Note that $H\le_fG$ here stands for finite-index subgroups $H$. Other zeta functions are obtained by restricting to certain types of subgroups, like normal ones for example. If $G$ is nilpotent then it is a direct product of Sylow $p$-subgroups, which yields an Euler product factorization for $\zeta_G(s)$. This allows a handful of examples to be explicitly computed with the Riemann zeta function; e.g.:
$$\zeta_{\Bbb Z^d}(s)=\zeta(s)\zeta(s-1)\cdots\zeta(s-d+1);\tag{3}$$ $$\zeta_{H_{\large3}(\Bbb Z)}(s)=\frac{\zeta(s)\zeta(s-1)\zeta(2s-2)\zeta(2s-3)}{\zeta(3s-3)},\tag{4}$$
where $H_3(\Bbb Z)$ is the discrete Heisenberg group. See $p$-adic integration and the theory of groups (.ps) or I bet it would also be somewhere in Lectures on Profinite Topics in Group Theory's Chapter $\rm III$.
Professor B. Sury points out the curious resemblance with a well-known identity of Ramanujan:
$$\sum_{n\ge1}\frac{\sigma_a(n)\sigma_b(n)}{n^s}=\frac{\zeta(s)\zeta(s-a)\zeta(s-b)\zeta(s-a-b)}{\zeta(2s-a-b)}.\tag{5}$$
Can we see $(5)$ as the zeta functions associated to a 'natural' family of groups indexed by $a,b\in\Bbb N$?
