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:

\(\ds \map \zeta s\)

\(=\)

\(\ds \prod_p \frac 1 {1 - p^{-s} }\)

\(\ds \leadsto \ \ \)

\(\ds \log \map \zeta s\)

\(=\)

\(\ds \log \prod_p \frac 1 {1 - p^{-s} }\)

\(\ds \)

\(=\)

\(\ds \sum_p \map \log {\frac 1 {1 - p^{-s} } }\)

Sum of Logarithms

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \paren {\map \pi n - \map \pi {n - 1} } \map \log {\frac 1 {1 - n^{-s} } }\)

Definition of Prime-Counting Function

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \map \pi n \paren {\map \log {\frac 1 {1 - n^{-s} } } - \map \log {\frac 1 {1 - \paren {n + 1}^{-s} } } }\)

as $\map \pi {-1} = 0$

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \map \pi n \paren {\map \log {1 - \paren {n + 1}^{-s} } - \map \log {1 - n^{-s} } }\)

Logarithm of Power

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

\(\ds \frac \d {\d x} \map \log {1 - x^{-s} }\)

\(=\)

\(\ds \frac {s x^{- s - 1} } {1 - x^{-s} }\)

\(\ds \)

\(=\)

\(\ds \frac s {x \paren {x^s - 1} }\)

Hence:

\(\ds \log \map \zeta s\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \map \pi n \int_n^{n + 1} \frac s {x \paren {x^s - 1} } \rd x\)

Fundamental Theorem of Calculus

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty s \int_n^{n + 1} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x\)

\(\ds \)

\(=\)

\(\ds s \int_0^{\mathop \to \infty} \frac {\map \pi x} {x \paren {x^s - 1} } \rd x\)

$\blacksquare$