Formula:DLMF:25.5:E6

$\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} \right) \frac{x^{s-1}}{\expe^x} \diff{x} }$

Constraint(s)

$\displaystyle {\displaystyle \realpart{s} > -1}$ &
${\displaystyle {\displaystyle s\neq 1}}$

Proof

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Comes from

$\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }$
by using the identity $\displaystyle {\displaystyle \expe^{-x} = (1-\expe^{-x})/(\expe^x-1)}$
in the integral $\displaystyle {\displaystyle \EulerGamma@{s} = \int_0^{\infty} \expe^{-x} x^{s-1} \diff{x}}$
(see
$\displaystyle {\displaystyle \EulerGamma@{z} = \int_0^\infty \expe^{-t} t^{z-1} \diff{t} }$
) together with

$\displaystyle {\displaystyle \EulerGamma@{z+1} = z \EulerGamma@{z} }$ .