Properties of $\zeta(s)\zeta(2s)\zeta(3s)...$ Let's consider the Dirichlet series $f(s)=\sum_{n=1}^\infty a_n n^{-s}$, where $a_n$ is the number of non-isomorphic abelian groups of order $n$. Now $a_n$ is weakly multiplicative and $a_{p^k}=P(k)=$ partition number of $k$, so we get $f(s)=\prod_{p} \sum_{k=0}^\infty P(k) p^{-ks}=\prod_{p} \prod_{k=1}^\infty \frac{1}{1-p^{-ks}}$ 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.
 A: 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 non-isomorphism 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
$$W(s)=(\Gamma(s/2)^{-1}\Gamma_2(s))^{1/2}\pi^{s^2/4}2^{(s-1)(s-2)/4}\zeta_\infty(s)$$
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.
