Periodic Zeta Function

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Periodic Zeta Function

\PeriodicZeta @ x s = n = 1 e 2 π i n x n s \PeriodicZeta @ 𝑥 𝑠 superscript subscript 𝑛 1 2 imaginary-unit 𝑛 𝑥 superscript 𝑛 𝑠 {\displaystyle{\displaystyle{\displaystyle\PeriodicZeta@{x}{s}=\sum_{n=1}^{% \infty}\frac{{\mathrm{e}^{2\pi\mathrm{i}nx}}}{n^{s}}}}} {\displaystyle \PeriodicZeta@{x}{s} = \sum_{n=1}^\infty \frac{\expe^{2 \cpi \iunit n x}}{n^s} }

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


\PeriodicZeta @ x s = Γ ( 1 - s ) ( 2 π ) 1 - s ( e π i ( 1 - s ) / 2 \HurwitzZeta @ 1 - s x + e π i ( s - 1 ) / 2 \HurwitzZeta @ 1 - s 1 - x ) \PeriodicZeta @ 𝑥 𝑠 Euler-Gamma 1 𝑠 superscript 2 1 𝑠 imaginary-unit 1 𝑠 2 \HurwitzZeta @ 1 𝑠 𝑥 imaginary-unit 𝑠 1 2 \HurwitzZeta @ 1 𝑠 1 𝑥 {\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)}}} {\displaystyle \PeriodicZeta@{x}{s} = \frac{\EulerGamma@{1-s}}{(2 \cpi)^{1-s}} \* \left( \expe^{\cpi \iunit (1-s)/2} \HurwitzZeta@{1-s}{x} + \expe^{\cpi \iunit (s-1)/2} \HurwitzZeta@{1-s}{1-x} \right) }

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


\HurwitzZeta @ 1 - s x = Γ ( s ) ( 2 π ) s ( e - π i s / 2 \PeriodicZeta @ x s + e π i s / 2 \PeriodicZeta @ - x s ) \HurwitzZeta @ 1 𝑠 𝑥 Euler-Gamma 𝑠 superscript 2 𝑠 imaginary-unit 𝑠 2 \PeriodicZeta @ 𝑥 𝑠 imaginary-unit 𝑠 2 \PeriodicZeta @ 𝑥 𝑠 {\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)}}} {\displaystyle \HurwitzZeta@{1-s}{x} = \frac{\EulerGamma@{s}}{(2 \cpi)^s} \left( \expe^{-\cpi \iunit s/2} \PeriodicZeta@{x}{s} + \expe^{\cpi \iunit s/2} \PeriodicZeta@{-x}{s} \right) }

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