i have posted this question on MO, and they referred me to post here . one starts with the formal definition of zeta :
$\displaystyle \zeta (s)=\prod_{p}\frac{1}{1-p^{-s}} $
then : $ \ln(\zeta (s))= -\sum_{p}\ln(1-p^{-s})=\sum_{p}\sum_{n=1}^{\infty}\frac{p^{-sn}}{n}$
using the trick : $\displaystyle p^{-sn}=s\int_{p^{n}}^{\infty}x^{-s-1}dx $
then :
$ \frac{\ln\zeta (s)}{s} = \sum_{p}\sum_{n=1}^{\infty}\int_{p^{n}}^{\infty}x^{-s-1}dx$
up until now, things make perfect sense , but the following line is mysterious to me :
$ \frac{\ln\zeta(s)}{s}=\int_{0}^{\infty}f(x)x^{-s-1}dx $
where $f(x) $ is the weighted-prime counting function . how is this formula derived !?!?