Periodic Zeta Function

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Periodic Zeta Function

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Periodic Zeta Function

${\displaystyle{\displaystyle{\displaystyle\PeriodicZeta@{x}{s}=\sum_{n=1}^{\infty}\frac{{\mathrm{e}^{2\pi\mathrm{i}nx}}}{n^{s}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle x}}}$ real &
${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ if ${\displaystyle{\displaystyle{\displaystyle x}}}$ is an integer, ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ otherwise

${\displaystyle{\displaystyle{\displaystyle\PeriodicZeta@{x}{s}=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left({\mathrm{e}^{\pi\mathrm{i}(1-s)/2}}\HurwitzZeta@{1-s}{x}+{\mathrm{e}^{\pi\mathrm{i}(s-1)/2}}\HurwitzZeta@{1-s}{1-x}\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{1-s}{x}=\frac{\Gamma\left(s\right)}{(2\pi)^{s}}\left({\mathrm{e}^{-\pi\mathrm{i}s/2}}\PeriodicZeta@{x}{s}+{\mathrm{e}^{\pi\mathrm{i}s/2}}\PeriodicZeta@{-x}{s}\right)}}}$

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

<< Polylogarithms

Periodic Zeta Function

Lerch's Transcendent >>