§25.13 Periodic Zeta Function

ⓘ

Keywords:: Hurwitz zeta function, periodic zeta function, polylogarithms, relation to Hurwitz zeta function, relation to polylogarithms, relations to other functions
Notes:: See Apostol (1976, Section 12.7).
Permalink:: http://dlmf.nist.gov/25.13
See also:: Annotations for Ch.25

The notation $F\left(x,s\right)$ is used for the polylogarithm $\operatorname{Li}_{s}\left(e^{2\pi ix}\right)$ 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		$F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$
$0<x<1$ , $\Re s>1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $F\left(\NVar{x},\NVar{s}\right)$ : periodic zeta function, $\Re$ : real part, $x$ : real variable and $s$ : complex variable Keywords: connection formula Source: Apostol (1976, Exercise 2, p. 273) Permalink: http://dlmf.nist.gov/25.13.E2 Encodings: TeX, pMML, png See also: Annotations for §25.13 and Ch.25

25.13.3		$\zeta\left(1-s,x\right)=\frac{\Gamma\left(s\right)}{(2\pi)^{s}}\left(e^{-\pi is% /2}F\left(x,s\right)+e^{\pi is/2}F\left(-x,s\right)\right),$
$\Re s>0$ if $0<x<1$ ; $\Re s>1$ if $x=1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $F\left(\NVar{x},\NVar{s}\right)$ : periodic zeta function, $\Re$ : real part, $x$ : real variable and $s$ : complex variable Keywords: connection formula Source: Apostol (1976, (10), p. 257) Notes: This formula is equivalent to (25.11.9). Referenced by: Erratum (V1.1.4) for Equation (25.13.3) Permalink: http://dlmf.nist.gov/25.13.E3 Encodings: TeX, pMML, png See also: Annotations for §25.13 and Ch.25