Integral Representation of Riemann Zeta Function in terms of Gamma Function

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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
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 1}^\infty \int_0^\infty t^{s - 1} e^{-nt} \rd t\) 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 $n t \to t$
\(\displaystyle \) \(=\) \(\displaystyle \map \zeta s \map \Gamma s\)

$\blacksquare$


Sources