§25.13 Periodic Zeta Function

Notes:: See Apostol (1976, pp. 257 and 273).
Keywords:: Hurwitz zeta function, periodic zeta function, polylogarithms
Permalink:: http://dlmf.nist.gov/25.13

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:: $\mathop{F\/}\nolimits\!\left(x,s\right)$ : periodic zeta function
Symbols:: $e$ : 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

where $\realpart{s}>1$ if $x$ is an integer, $\realpart{s}>0$ 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 $\mathop{F\/}\nolimits\!\left(x,s\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-s\right)}{(2\pi)^{{1-s}}}\*\left(e^{{\pi i(1-s)/2}}\mathop{\zeta\/}\nolimits\!\left(1-s,x\right)+e^{{\pi i(s-1)/2}}\mathop{\zeta\/}\nolimits\!\left(1-s,1-x\right)\right),$ $0<x<1$ , $\realpart{s}>1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $e$ : base of exponential function, $\mathop{F\/}\nolimits\!\left(x,s\right)$ : periodic zeta function, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.13.E2
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ : Hurwitz zeta function, $e$ : base of exponential function, $\mathop{F\/}\nolimits\!\left(x,s\right)$ : periodic zeta function, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.13.E3
Encodings:: TeX, pMML, png