Riemann Zeta Function and Prime Counting Function

Theorem
For $\map \Re s > 1$:


 * $\ds \log \map \zeta s = s \int_0^{\mathop \to \infty} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x$

where:
 * $\zeta$ denotes the Riemann Zeta Function
 * $\pi$ 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: