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 that x is $x*zeta(2)+O(x^(3/4))$ and then expand that to the product of 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 $xzeta(2)$ but I can't find a contour that converges or the exact error term. 

I have also tried to use the hyperbola method and playing with them as cauchy sequences but that also doesn't seem to work. 

If anyone could help me I would truly apreciate it.