# §25.8 Sums

 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 $\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(nk\right)-1)=\ln\left(\prod_{% j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)\right),$ $n=2,3,4,\dots$.
 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).