I have this $a(n)$ is unknown multiplicative function and $b(n)=n$. Let $\zeta(x)$ be Riemann zeta function. And $$B(x)=\zeta^2(x)A(x).$$ where $B(x)=\sum_{n\in \mathbb{N}}\frac{b(n)}{n^x}$ (same for $A(x)$).
I wrote both sides as Euler products, ea. $$B(x)=\prod_p(1+\frac{b(p)}{p^x}+\frac{b(p^2)}{p^{2x}}+\cdots)$$ and $\zeta^2(x)=\prod_p\frac{p^{2x}}{(p^x-1)^2}=\prod_p(1+\frac{2}{p^x-1}+\frac{1}{(p^x-1)^2})$ and I thought I should compare what is next to $1/p^{kx}$ in "p parthenses", after I multiply on the right side, to find $a(p^{kx})$ but denumerator in this zeta function expression makes problem since it's not in that form. I know I can write. Any ideas?
If we use $$A(x) = B(x) \zeta^{-2}(x),$$ The Euler product of the right side is $$ \prod_p \left( 1 + \frac p{p^x} + \frac{ p^2}{p^{2x}} + \cdots\right) \left( 1 - \frac 1{p^x}\right)^2.$$ Now, try expanding the right side.