Proof of an identity concerning the prime $\zeta$ function

Question

I have to prove the following identity: let $P(s)=\sum_p\frac{1}{p^s}$, for $Re(s)>1$, then \begin{equation} P(s)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log(\zeta(ns)). \end{equation} I proved that \begin{equation} \log(\zeta(s))=\sum_{n=1}^{\infty}\frac{P(ns)}{n}. \end{equation} I know I have to use Möbius inversion formula to deduce the result from this identity, but I can't see how.

Answer 1

A more general form of Möbius inversion is the following: $$g(s) = \sum_{n=1}^\infty \frac{f(ns)}n \iff f(s) = \sum_{n=1}^\infty \mu(n)\,\frac{g(ns)}{n},$$ which is exactly what you need.