I have the following question: Let $F(s)$ be the Dirichlet series associated to $$f(n)= \sum_{d\mid n} \frac{\log(d)}{d} $$. My answer has to depend on the zeta function. (i.e simplify F(s) so that we can see its relation to the zeta function) Here is what I have so far:
$$ F(s)= \sum_{n=1}^{infinity} \sum_{d\mid n} \frac{\log(d)}{d} \frac{1}{n^s}$$ $$=\prod_{P} (1+ \frac{f(P)}{P^s} + \frac{f(P^2)}{P^{2s}}+....+)$$ $$ = 1+ \frac{\log P}{P *P^s}+ \frac{\log(P)}{P P^{2s}}+ \frac{\log(P^2)}{P^2 P^{2s}}+.... $$
I am stuck. I don't know how to write the sum in function of $$\zeta(s)= \prod_{P} (1+ 1/P^s + 1/P^{2s}....)$$
thank you!