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:

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

where $\Gamma$ is the Gamma function.


Corollary

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

$\ds \map \zeta s = \frac 1 {\map \Gamma s} \int_0^1 \frac {\paren {\map \ln {\dfrac 1 u} }^{s - 1} } {1 - u} \rd u$

where $\Gamma$ is the Gamma function.


Proof

\(\ds \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \rd t\) \(=\) \(\ds \int_0^\infty \frac {t^{s - 1} } {e^t - 1} \times \dfrac {e^{-t} } {e^{-t} }\rd t\) multiplying by 1
\(\ds \) \(=\) \(\ds \int_0^\infty \frac {t^{s - 1} e^{-t} } {1 - e^{-t} } \rd t\)
\(\ds \) \(=\) \(\ds \int_0^\infty t^{s - 1} e^{-t} \sum_{n \mathop = 0}^\infty e^{-n t} \rd t\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds \int_0^\infty t^{s - 1} \sum_{n \mathop = 0}^\infty e^{-\paren {n + 1} t} \rd t\) Product of Powers
\(\ds \) \(=\) \(\ds \int_0^\infty t^{s - 1} \sum_{n \mathop = 1}^\infty e^{-n t} \rd t\) reindexing the sum
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \int_0^\infty t^{s - 1} e^{-n t} \rd t\) interchange of sum and integral is valid by Fubini's Theorem
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \int_0^\infty \paren {\dfrac u n}^{s - 1} e^{-u} \dfrac 1 n \rd u\) substituting $n t \to u$; $t \to \dfrac u n$ and $\rd t \to \dfrac 1 n \rd u$
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^s} \int_0^\infty u^{s - 1} e^{-u} \rd u\)
\(\ds \) \(=\) \(\ds \map \zeta s \map \Gamma s\) Definition of Riemann Zeta Function and Definition of Integral Form of Gamma Function

$\blacksquare$


Sources

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