Riemann Zeta Function and Prime Counting Function

Theorem
For $\operatorname{Re} \left({s}\right) > 1 $:


 * $\displaystyle \log \zeta \left({s}\right) = s \int_0^{\to \infty} \frac {\pi \left({x}\right)} {x \left({x^s - 1}\right)} \mathrm d x$

where $\zeta$ denotes the Riemann Zeta Function and $\pi$ denotes the Prime-Counting Function.

Proof
From the definition of the Riemann Zeta Function:

By Derivative of Logarithm Function and the Chain Rule for Derivatives:

Hence: