# §25.13 Periodic Zeta Function

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

 25.13.1 $F\left(x,s\right)\equiv\sum_{n=1}^{\infty}\frac{e^{2\pi inx}}{n^{s}},$ ⓘ Defines: $F\left(\NVar{x},\NVar{s}\right)$: periodic zeta function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: definition, infinite series, series representation Source: Apostol (1976, (9), p. 257) Permalink: http://dlmf.nist.gov/25.13.E1 Encodings: TeX, pMML, png See also: Annotations for §25.13 and Ch.25

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

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

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