§25.8 Sums

25.8.1		$$\sum _{k=2}^{\mathrm{\infty }}\left(\zeta \left(k\right)-1\right)=1.$$
ⓘ Symbols: $\zeta \left(s\right)$: 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		$$\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)}{(k+1)!}\left(\zeta \left(s+k\right)-1\right)=\mathrm{\Gamma }\left(s-1\right),$$
$s\ne 1,0,-1,-2,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: 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) Referenced by: §25.8 Permalink: http://dlmf.nist.gov/25.8.E2 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

25.8.3		$$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_k\zeta \left(s+k\right)}{k!2^{s+k}}=(1-2^{-s})\zeta \left(s\right),$$
$s\ne 1$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: 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), §25.8, 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 ${\sum }_{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)\zeta \left(s+k\right)}{k!\mathrm{\Gamma }\left(s\right)2^{s+k}}$ more concisely as ${\sum }_{k=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_k\zeta \left(s+k\right)}{k!2^{s+k}}$ using the Pochhammer symbol. See also: Annotations for §25.8 and Ch.25

25.8.4		$$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}(\zeta \left(nk\right)-1)=\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(2-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$
$n=2,3,4,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: 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, $\ln z$: 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		${\displaystyle \sum _{k=2}^{\mathrm{\infty }}}\zeta \left(k\right)z^k$	$=-\gamma z-z\psi \left(1-z\right),$
.
ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable Keywords: infinite series Source: Erdélyi et al. (1953a, (5), p. 45); with $z\mapsto -z$ Referenced by: (25.8.7), §25.8 Permalink: http://dlmf.nist.gov/25.8.E5 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25
25.8.6		${\displaystyle \sum _{k=0}^{\mathrm{\infty }}}\zeta \left(2k\right)z^{2k}$	$=-\frac{1}{2}\pi z\cot \left(\pi z\right),$
.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $k$: nonnegative integer and $z$: complex variable Keywords: infinite series Source: Erdélyi et al. (1953a, (3), p. 51); with $z\mapsto \pi z$ Referenced by: (25.8.8), §25.8 Permalink: http://dlmf.nist.gov/25.8.E6 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25
25.8.7		${\displaystyle \sum _{k=2}^{\mathrm{\infty }}}{\displaystyle \frac{\zeta \left(k\right)}{k}}{z}^k$	$=-\gamma z+\ln \mathrm{\Gamma }\left(1-z\right),$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable Keywords: infinite series Source: Derivable from (25.8.5) by dividing by $z$ and integrating. Referenced by: §25.8 Permalink: http://dlmf.nist.gov/25.8.E7 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25
25.8.8		${\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{\zeta \left(2k\right)}{k}}{z}^{2k}$	$=\ln \left({\displaystyle \frac{\pi z}{\sin \left(\pi z\right)}}\right),$
.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\sin z$: sine function, $k$: nonnegative integer and $z$: complex variable Keywords: infinite series Source: Derivable from (25.8.6) by dividing by $z$ and integrating. Referenced by: §25.8 Permalink: http://dlmf.nist.gov/25.8.E8 Encodings: TeX, pMML, png See also: Annotations for §25.8 and Ch.25

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

25.8.10		$$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4{\pi }^2}\zeta \left(3\right).$$
ⓘ Symbols: $\zeta \left(s\right)$: 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) Referenced by: §25.8 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).