Formula:DLMF:25.5:E2

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ζ ( s ) = 1 Γ ( s + 1 ) 0 x x s ( x - 1 ) 2 x Riemann-zeta s 1 Euler-Gamma s 1 superscript subscript 0 superscript x superscript x s superscript superscript x 1 2 x {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{\mathop{% \Gamma\/}\nolimits\!\left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}% -1)^{2}}dx}


Constraint(s)


s > 1 𝑠 1 {\displaystyle\realpart{s}>1}


Proof


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Integrate
ζ ( s ) = 1 Γ ( s ) 0 x s - 1 x - 1 x Riemann-zeta s 1 Euler-Gamma s superscript subscript 0 superscript x s 1 superscript x 1 x {\displaystyle\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{\mathop{% \Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}dx}
by parts.



Symbols List


ζ 𝜁 {\displaystyle\zeta}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
Γ Γ {\displaystyle\Gamma}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
{\displaystyle\int}  : integral : http://dlmf.nist.gov/1.4#iv
{\displaystyle e}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
a b 𝑏 𝑎 {\displaystyle\diff[a]{b}}  : differential : http://dlmf.nist.gov/1.4#iv
a 𝑎 {\displaystyle\Re{\,\hskip 0.0pta}}  : real part : http://dlmf.nist.gov/1.9#E2

Bibliography


Equation (1), Section 25.5 of DLMF.

URL links


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