# §25.13 Periodic Zeta Function

The notation $\mathop{F\/}\nolimits\!\left(x,s\right)$ is used for the polylogarithm $\mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(e^{2\pi ix}\right)$ with $x$ real:

 25.13.1 $\mathop{F\/}\nolimits\!\left(x,s\right)=\sum_{n=1}^{\infty}\frac{e^{2\pi inx}}% {n^{s}},$ Defines: $\mathop{F\/}\nolimits\!\left(\NVar{x},\NVar{s}\right)$: periodic zeta function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Permalink: http://dlmf.nist.gov/25.13.E1 Encodings: TeX, pMML, png See also: Annotations for 25.13

where $\Re{s}>1$ if $x$ is an integer, $\Re{s}>0$ otherwise.

$\mathop{F\/}\nolimits\!\left(x,s\right)$ is periodic in $x$ with period 1, and equals $\mathop{\zeta\/}\nolimits\!\left(s\right)$ when $x$ is an integer. Also,

 25.13.2 $\mathop{F\/}\nolimits\!\left(x,s\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(1-s)/2}\mathop{\zeta\/}% \nolimits\!\left(1-s,x\right)+e^{\pi i(s-1)/2}\mathop{\zeta\/}\nolimits\!\left% (1-s,1-x\right)\right),$ $0, $\Re{s}>1$,
 25.13.3 $\mathop{\zeta\/}\nolimits\!\left(1-s,x\right)=\frac{\mathop{\Gamma\/}\nolimits% \!\left(s\right)}{(2\pi)^{s}}\left(e^{-\pi is/2}\mathop{F\/}\nolimits\!\left(x% ,s\right)+e^{\pi is/2}\mathop{F\/}\nolimits\!\left(-x,s\right)\right),$ $0, $\Re{s}>0$.