Zeta function the the orime counting function

[mmzaj]

i have a question about the relation between the riemann zeta function and the prime counting function . one starts with the formal definition of zeta :
\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 :
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 ??

 

[mmzaj]

i have done a mistake
\frac{ln\zeta (s)}{s} = \sum_{p}\sum_{n=1}^{\infty}\frac{1}{n}\int_{p^{n}}^{\infty}x^{-s-1}dx

after some digging , the integral follows from the fact , that for any function g(x) :

\sum_{p}\int_{p^{n}}^{\infty}g(x)dx= \int_{0}^{\infty} \pi(x^{1/n}) g(x)dx

\pi(x) being the prime counting function .
but this line isn't so clear to me !