Integral Representation of Riemann Zeta Function in terms of Gamma Function/Corollary

Corollary to Integral Representation of Riemann Zeta Function in terms of Gamma Function

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.

$\ds \map \zeta {s}$

$=$

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

$\ds $

$=$

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

multiplying by 1

$\ds $

$=$

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

$\ds $

$=$

$\ds \frac 1 {\map \Gamma s} \int_1^0 \frac {\paren {\map \ln {\dfrac 1 u} }^{s - 1} } {1 - u} \paren {-\rd u }$

substituting $e^{-t} \to u$; $t \to \map \ln {\dfrac 1 u} $ and $-e^{-t}\rd t \to \rd u$

$\ds $

$=$

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

$\blacksquare$