Formula:DLMF:25.5:E6

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${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{{\mathrm{e}^{x}}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}^{x}}}\mathrm{d}x}}}$

Constraint(s)

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

Proof

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Comes from

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{{\mathrm{e}^{x}}-1}\mathrm{d}x}}}$
by using the identity ${\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{-x}}=(1-{\mathrm{e}^{-x}})/({\mathrm{e}^{x}}-1)}}}$
in the integral ${\displaystyle{\displaystyle{\displaystyle\Gamma\left(s\right)=\int_{0}^{\infty}{\mathrm{e}^{-x}}x^{s-1}\mathrm{d}x}}}$
(see
${\displaystyle{\displaystyle{\displaystyle\Gamma\left(z\right)=\int_{0}^{\infty}{\mathrm{e}^{-t}}t^{z-1}\mathrm{d}t}}}$
) together with

${\displaystyle{\displaystyle{\displaystyle\Gamma\left(z+1\right)=z\Gamma\left(z\right)}}}$.

Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (6), Section 25.5 of DLMF.

URL links

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