I was looking for representations of $\log \zeta$ and found these two:
- $ \displaystyle \log\zeta(s)=\color{red}{s}\sum_{n>0} \frac{P(ns)}{n\color{red}{s}}$ from here [$\color{red}{s}$ inserted by me],
- $ \displaystyle \log \zeta(s) = s \int_0^\infty \frac{\pi(x)}{x(x^s-1)}\,dx, $ from there.
Identify $x$ with $n$, does this somehow imply: $ \frac{P(ns)}{\color{black}{s}}=\frac{\pi(n)}{n^s-1}\; . $ If so, how to prove it? (Disproven here ) If not, how are 1. and 2. related?