Let's consider the Dirichlet series $f(s)=\sum_{n=1}^\infty a_n n^{s}$, where $a_n$ is the number of nonisomorphic abelian groups of order $n$. Now $a_n$ is weakly multiplicative and $a_{p^k}=P(k)=$ partition number of $k$, so we get $f(s)=\prod_{p} \sum_{k=0}^\infty P(k) p^{ks}=\prod_{p} \prod_{k=1}^\infty \frac{1}{1p^{ks}}$ because of the generating function of the partition number. So we get $f(s)=\prod_{k=1}^\infty \zeta(k s)$ (where everything converges absolutely).
So my question is: what is known about this function? Is there a functional equation or an analytic continuation?
Thank you very much.

4$\begingroup$ It has a meromorphic continuation with simple poles at $1/k$ and a natural boundary on $\Re(s) = 0$ (nontrivial zeros $\rho$ of $\zeta(s)$ are copied at $\rho/k$) . $\endgroup$– reunsCommented Aug 16, 2017 at 9:57
1 Answer
As usual when counting objects, it is preferable to assign a weight proportional to the inverse of their automorphism group. While the function $\zeta(s)\zeta(2s)...$ does count the number of nonisomorphism abelian groups of order $n$, if you now count them with the factor $1/Aut(G)$ the function becomes $\zeta_\infty(s)=\zeta(s+1)\zeta(s+2)\zeta(s+3)...$ which is MUCH more interesting. What follows is directly taken from the basic paper I wrote with H.W. Lenstra on heuristics for class groups (Springer L.N. 1068), in the special case of abelian groups (one can do the same for finitely generated modules over Dedekind domains). Set $$W(s)=(\Gamma(s/2)^{1}\Gamma_2(s))^{1/2}\pi^{s^2/4}2^{(s1)(s2)/4}\zeta_\infty(s)$$ where $\Gamma_2$ is Barnes's double gamma function, and $$\Lambda(s)=W(s)W(s)\sin^2(\pi s)/(\pi^2C^2)$$ where $C$ is an easily given normalizing constant.
Then $\Lambda(s)$ extends to an entire function of order $2$ (like the Selberg zeta function, but with quite different properties), which is of course even, and its functional equation simply says that it is periodic of period $1$ (morally speaking, it is a gamma factor times $\prod_{n\in\Bbb Z}\zeta(s+n)$), this being equivalent to the functional equation of the zeta function itself.
What's fun about this function is that if you graph it on the real line (try it!) you get the constant $1$ (or some other constant if you don't know the value of $C$), which is impossible.
In fact, its graph is close to less than $2.10^{38}$ from the constant $1$, as an easy exercise (or a look at the paper) shows.
Just my 2 cents because it's fun.

$\begingroup$ Also $\prod_{n=0}^{N1} \zeta(s+n)$ has its own functional equation. It is the zeta function of the projective variety $ \mathbb{P}^N$ and it should be the Mellin transform of an automorphic Eisenstein series. $\endgroup$– reunsCommented Aug 17, 2017 at 3:46