25 Zeta and Related FunctionsRelated Functions25.12 Polylogarithms25.14 Lerch’s Transcendent

§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 $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real:

25.13.1		$$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$
ⓘ Defines: $F(x,s)$: 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 $\mathrm{\Re }s>1$ if $x$ is an integer, $\mathrm{\Re }s>0$ otherwise.

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

25.13.2		$$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$
, $\mathrm{\Re }s>1$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: 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, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: 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		$$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$
$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: 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, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: 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

25.12 Polylogarithms25.14 Lerch’s Transcendent

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