§25.8 Sums

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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 $\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1\right)=1.$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : 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 $\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(k+1)!}\left(\zeta\left(s+k% \right)-1\right)=\Gamma\left(s-1\right),$ $s\neq 1,0,-1,-2,\dots$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : factorial (as in ), : nonnegative integer and : 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 $\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\zeta\left(s+k\right)}{k!2^{s+k}}% =(1-2^{-s})\zeta\left(s\right),$ $s\neq 1$ .

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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 ), : nonnegative integer and : complex variable
Keywords:: infinite series
Source:: Srivastava (1988, (2.6), p. 131); with 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 $\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(nk\right)-1)=\ln\left(\prod_{% j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)\right),$ $n=2,3,4,\dots$ .

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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, : nonnegative integer and : 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 $\sum_{k=2}^{\infty}\zeta\left(k\right)z^{k}=-\gamma z-z\psi\left(1-z\right),$

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Symbols:: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : nonnegative integer and : 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 $\sum_{k=0}^{\infty}\zeta\left(2k\right)z^{2k}=-\tfrac{1}{2}\pi z\cot\left(\pi z% \right),$

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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, : nonnegative integer and : 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 $\sum_{k=2}^{\infty}\frac{\zeta\left(k\right)}{k}z^{k}=-\gamma z+\ln\Gamma\left% (1-z\right),$

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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, : nonnegative integer and : complex variable
Keywords:: infinite series
Proof sketch:: Derivable from (25.8.5) by dividing by 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 $\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z^{2k}=\ln\left(\frac{\pi z}{% \sin\left(\pi z\right)}\right),$

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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, : nonnegative integer and : complex variable
Keywords:: infinite series
Proof sketch:: Derivable from (25.8.6) by dividing by 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 $\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac% {1}{2}\ln 2.$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\ln\NVar{z}$ : principal branch of logarithm function and : 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 $\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}% -\frac{7}{4\pi^{2}}\zeta\left(3\right).$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter and : 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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