25 Zeta and Related Functions Riemann Zeta Function 25.7 Integrals 25.9 Asymptotic Approximations

§25.8 Sums

ⓘ

Keywords:: Riemann zeta function, sums
Notes:: See Adamchik and Srivastava (1998), Srivastava (1988), and Erdélyi et al. (1953a, Section 1.17).
Permalink:: http://dlmf.nist.gov/25.8
See also:: Annotations for Ch.25

25.8.1		$\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1\right)=1.$
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and $k$ : nonnegative integer Keywords: infinite series Source: Adamchik and Srivastava (1998, (1.4), p. 132) Permalink: http://dlmf.nist.gov/25.8.E1 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

25.8.2		$\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(k+1)!}\left(\zeta\left(s+k% \right)-1\right)=\Gamma\left(s-1\right),$
$s\neq 1,0,-1,-2,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $s$ : complex variable Keywords: infinite series Source: Landau (1953, (2), p. 273); with (5.2.4), (5.2.5) Permalink: http://dlmf.nist.gov/25.8.E2 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

25.8.3		$\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\zeta\left(s+k\right)}{k!2^{s+k}}% =(1-2^{-s})\zeta\left(s\right),$
$s\neq 1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $s$ : complex variable Keywords: infinite series Source: Srivastava (1988, (2.6), p. 131); with $k=0$ term Referenced by: §25.11(iv), Erratum (V1.0.9) for Chapters 7, 25 Permalink: http://dlmf.nist.gov/25.8.E3 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): We have rewritten the original summation $\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)\zeta\left(s+k\right)}{k!\Gamma% \left(s\right)2^{s+k}}$ more concisely as $\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\zeta\left(s+k\right)}{k!2^{s+k}}$ using the Pochhammer symbol. See also: Annotations for §25.8 and Ch.25

25.8.4		$\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(nk\right)-1)=\ln\left(\prod_{% j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)\right),$
$n=2,3,4,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $n$ : nonnegative integer Keywords: infinite series Source: Adamchik and Srivastava (1998, (2.7), p. 135) Permalink: http://dlmf.nist.gov/25.8.E4 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

25.8.5		$\displaystyle\sum_{k=2}^{\infty}\zeta\left(k\right)z^{k}$	$\displaystyle=-\gamma z-z\psi\left(1-z\right),$
$\|z\|<1$ .
ⓘ Symbols: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $k$ : nonnegative integer and $z$ : complex variable Keywords: infinite series Source: Erdélyi et al. (1953a, (1.17.5), p. 45); with $z\mapsto-z$ Referenced by: (25.8.7) Permalink: http://dlmf.nist.gov/25.8.E5 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25
25.8.6		$\displaystyle\sum_{k=0}^{\infty}\zeta\left(2k\right)z^{2k}$	$\displaystyle=-\tfrac{1}{2}\pi z\cot\left(\pi z\right),$
$\|z\|<1$ .
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $k$ : nonnegative integer and $z$ : complex variable Keywords: infinite series Source: Erdélyi et al. (1953a, (1.20.3), p. 51); with $z\mapsto\pi z$ Referenced by: (25.8.8) Permalink: http://dlmf.nist.gov/25.8.E6 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25
25.8.7		$\displaystyle\sum_{k=2}^{\infty}\frac{\zeta\left(k\right)}{k}z^{k}$	$\displaystyle=-\gamma z+\ln\Gamma\left(1-z\right),$
$\|z\|<1$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $z$ : complex variable Keywords: infinite series Proof sketch: Derivable from (25.8.5) by dividing by $z$ and integrating. Permalink: http://dlmf.nist.gov/25.8.E7 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25
25.8.8		$\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z^{2k}$	$\displaystyle=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right),$
$\|z\|<1$ .
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer and $z$ : complex variable Keywords: infinite series Proof sketch: Derivable from (25.8.6) by dividing by $z$ and integrating. Permalink: http://dlmf.nist.gov/25.8.E8 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

25.8.9		$\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac% {1}{2}\ln 2.$
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\ln\NVar{z}$ : principal branch of logarithm function and $k$ : nonnegative integer Keywords: infinite series Source: Srivastava and Choi (2001, (467), p. 212) Permalink: http://dlmf.nist.gov/25.8.E9 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

25.8.10		$\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}% -\frac{7}{4\pi^{2}}\zeta\left(3\right).$
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter and $k$ : nonnegative integer Keywords: infinite series Source: Ewell (1990, (1), p. 219) Permalink: http://dlmf.nist.gov/25.8.E10 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

For other sums see Prudnikov et al. (1986b, pp. 648–649), Hansen (1975, pp. 355–357), Ogreid and Osland (1998), and Srivastava and Choi (2001, Chapter 3).

25.7 Integrals 25.9 Asymptotic Approximations

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