# Zeta functions of groups and an identity of Ramanujan

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 $a,b\in\Bbb N$?

• Upon reflection, I'm not sure there's anything special about this family of functions: the same question applies equally well to many other finite products of $L$-functions. The answer to the general question may require tools so sophisticated or theory too vast to explicate here or that we don't have at present, so this may at any rate turn out to be a novel test-case.
– anon
Sep 18, 2012 at 4:06