Integral Representations

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In Terms of Elementary Functions

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{\expe^{x}x^{s}}{(\expe^{x}-1)^{2}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{\expe^{x}+1}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{\expe^{x}x^{s}}{(\expe^{x}+1)^{2}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{x^{s+1}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle-1<\realpart{s}<0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{\expe^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{\expe^{x}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}\frac{\Gamma\left(s+2m-1\right)}{\Gamma\left(s\right)}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{\expe^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{B_{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{\expe^{x}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-(2n+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2(1-2^{-s})\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh x}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{x^{s}}{(\sinh x)^{2}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos\!\left(s\mathrm{arctan}x\right)}{(1+x^{2})^{s/2}\cosh\!\left(\frac{1}{2}\cpi x\right)}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin\!\left(s\mathrm{arctan}x\right)}{(1+x^{2})^{s/2}(\expe^{2\cpi x}-1)}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin\!\left(s\mathrm{arctan}x\right)}{(1+x^{2})^{s/2}(\expe^{\cpi x}+1)}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

In Terms of Other Functions

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\cpi^{s/2}}{s(s-1)\Gamma\left(\frac{1}{2}s\right)}+\frac{\cpi^{s/2}}{\Gamma\left(\frac{1}{2}s\right)}\*\int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\diffd x}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\displaystyle\omega(x)=\sum_{n=1}^{\infty}\expe^{-n^{2}\cpi x}=\frac{1}{2}\left(\theta_{3}\!\left(0\middle|\iunit x\right)-1\right)}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\!\left(\cpi s\right)}{\cpi}\*\int_{0}^{\infty}(\ln\!\left(1+x\right)-\psi\left(1+x\right))x^{-s}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<\realpart{s}<1}}}$

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{s-1}+\frac{\sin@{\cpi s}}{\cpi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\digamma^{\prime}@{1+x}\right)x^{1-s}\diff{x}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<\realpart{s}<2}}}$, ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\!\left(\cpi s\right)}{\cpi}\int_{0}^{\infty}\left(\EulerConstant+\psi\left(1+x\right)\right)x^{-s-1}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<\realpart{s}<1}}}$

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{1+s}=\frac{\sin@{\cpi s}}{\cpi s}\int_{0}^{\infty}\digamma^{\prime}@{1+x}x^{-s}\diff{x}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<\realpart{s}<1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(s\right)\sin\!\left(\cpi s\right)}{\cpi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{\psi^{(m)}}\left(1+x\right)x^{-s}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle m=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<\realpart{s}<1}}}$

Contour Integrals

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\cpi\iunit}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{\expe^{-z}-1}\diffd z}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1,2,\dots}}}$ &

The integration contour is a loop around the negative real axis; it starts at ${\displaystyle{\displaystyle{\displaystyle-\infty}}}$, encircles the origin once in the positive direction without enclosing any of the points

${\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi i}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 4\pi i,\ldots,}}}$ and returns to ${\displaystyle{\displaystyle{\displaystyle-\infty}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{2\cpi\iunit(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{\expe^{-z}+1}\diffd z}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1,2,\dots}}}$ &

The contour here is any loop that encircles the origin in the positive direction

not enclosing any of the points ${\displaystyle{\displaystyle{\displaystyle\pm\pi i}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 3\pi i,\ldots}}}$

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