I have been looking at this for days and I am going insane. 

I need to show that for a Dirichlet series equal to $\zeta(s)\zeta(2s)$, the sum of the coefficients less than $x$ is $x\zeta(2)+O(x^{(3/4)})$, and then expand that to the $\Pi \zeta(ks)$ for all $k$ in an effort to find the formula for the number of non-isomorphic abelian groups.

I know that using Perron's formula there is a simple pole at $s=1$ that gives a residue of $X\zeta(2)$, but I can't find a contour that converges or the exact error term.