# Integral Representation of Riemann Zeta Function in terms of Gamma Function

## Theorem

For $\Re \paren s > 1$, the Riemann Zeta function is given by:

$\displaystyle \map \zeta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {t^{s - 1}} {e^t - 1} \rd t$

where $\Gamma$ is the Gamma function.

## Proof

 $\displaystyle \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \rd t$ $=$ $\displaystyle \int_0^\infty \frac {t^{s - 1} e^{-t} } {1 - e^{-t} } \rd t$ $\displaystyle$ $=$ $\displaystyle \int_0^\infty t^{s - 1} e^{-t} \sum_{n \mathop = 0}^\infty e^{-n t} \rd t$ Sum of Infinite Geometric Progression $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \int_0^\infty t^{s - 1} e^{-n t} \rd t$ interchange of sum and integral is valid by Fubini's Theorem $\displaystyle$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^s} \int_0^\infty t^{s - 1} e^{-t} \rd t$ substituting $n t \to t$ $\displaystyle$ $=$ $\displaystyle \map \zeta s \, \map \Gamma s$

$\blacksquare$