# 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:

 $\displaystyle \zeta \left({s}\right)$ $=$ $\displaystyle \prod_p \frac 1 {1 - p^{-s} }$ $\displaystyle \implies \ \$ $\displaystyle \log \zeta \left({s}\right)$ $=$ $\displaystyle \log \prod_p \frac 1 {1 - p^{-s} }$ $\displaystyle$ $=$ $\displaystyle \sum_p \log \left({\frac 1 {1 - p^{-s} } }\right)$ Sum of Logarithms $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \left({\pi \left({n}\right) - \pi \left({n - 1}\right)}\right) \log \left( \frac 1 {1 - n^{-s} } \right)$ by definition of Prime-Counting Function $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \pi \left({n}\right) \left({\log \left( \frac 1 {1 - n^{-s} } \right) - \log \left( \frac 1 {1 - \left({n + 1}\right)^{-s} }\right)}\right)$ as $\pi \left({-1}\right) = 0$ $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \pi \left({n}\right) \left({\log \left(1 - \left({n + 1}\right)^{-s}\right) - \log \left(1 - n^{-s} \right)}\right)$ Logarithms of Powers
 $\displaystyle \frac {\mathrm d } {\mathrm d x} \log \left({1 - x^{-s} }\right)$ $=$ $\displaystyle \frac {s x^{- s - 1} } {1 - x^{-s} }$ $\displaystyle$ $=$ $\displaystyle \frac s {x \left({x^s - 1}\right)}$

Hence:

 $\displaystyle \log \zeta \left({s}\right)$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \pi \left({n}\right) \int_n^{n + 1} \frac s {x \left({x^s - 1}\right)} \mathrm d x$ by Fundamental Theorem of Calculus $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty s \int_n^{n + 1} \frac {\pi \left({x}\right)} {x \left({x^s - 1}\right)} \mathrm d x$ $\displaystyle$ $=$ $\displaystyle s \int_0^{\to \infty} \frac {\pi \left({x}\right)} {x \left({x^s - 1}\right)} \mathrm d x$

$\blacksquare$