I can not find anywhere on the internet a proof of this formula :
$$\sum_{p\in\mathbb{P}, \; m\geq 1, \; p^m < x} \ln p = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac{1}{2}\ln \left(1-\frac{1}{x^2}\right) $$ where $\rho$ are non trivial zeros of the Riemann zeta funciton.
My starting point is the identity :
$$\sum_{p\in\mathbb{P},\; n\geq 1}p^{-ns}\ln p = -\ln 2\pi +\frac{s}{s-1} -\sum_{\rho\in Z}\frac{s}{\rho(s-\rho)}$$
If I set an $x$ and multiply the identity by $ \displaystyle \frac{x^s}{s}$ then we have :
$$\sum_{p\in\mathbb{P},\; n\geq 1} \frac{x^s}{sp^{ns}}\ln p = - \frac{x^s}{s} \ln 2\pi + \frac{x^s}{s-1} -\sum_{\rho\in Z}\frac{x^s}{\rho(s-\rho)}$$
and I was told to use the residues theorem and integrate over a "good" set.
I'm still not able to finish this, can someone eplain it to me or give me a link ?
Thank you.
You can find a proof of the explicit formula here.