# §25.8 Sums

 25.8.1 $\sum_{k=2}^{\infty}\left(\mathop{\zeta\/}\nolimits\!\left(k\right)-1\right)=1.$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function and $k$: nonnegative integer Permalink: http://dlmf.nist.gov/25.8.E1 Encodings: TeX, pMML, png See also: Annotations for 25.8
 25.8.2 $\sum_{k=0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+k\right)}{(k+1)!}% \left(\mathop{\zeta\/}\nolimits\!\left(s+k\right)-1\right)=\mathop{\Gamma\/}% \nolimits\!\left(s-1\right),$ $s\neq 1,0,-1,-2,\dots$.
 25.8.3 $\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\mathop{\zeta\/}\nolimits\!\left(% s+k\right)}{k!2^{s+k}}=(1-2^{-s})\mathop{\zeta\/}\nolimits\!\left(s\right),$ $s\neq 1$. Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $s$: complex variable Referenced by: §25.11(iv), §25.8, Other Changes 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 $\sum_{k=0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+k\right)\mathop{% \zeta\/}\nolimits\!\left(s+k\right)}{k!\mathop{\Gamma\/}\nolimits\!\left(s% \right)2^{s+k}}$ more concisely as $\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\mathop{\zeta\/}\nolimits\!\left(% s+k\right)}{k!2^{s+k}}$ using the Pochhammer symbol. Reported 2014-05-22 See also: Annotations for 25.8
 25.8.4 $\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\mathop{\zeta\/}\nolimits\!\left(nk% \right)-1)=\mathop{\ln\/}\nolimits\!\left(\prod_{j=0}^{n-1}\mathop{\Gamma\/}% \nolimits\!\left(2-e^{(2j+1)\pi i/n}\right)\right),$ $n=2,3,4,\dots$.
 25.8.5 $\displaystyle\sum_{k=2}^{\infty}\mathop{\zeta\/}\nolimits\!\left(k\right)z^{k}$ $\displaystyle=-\gamma z-z\mathop{\psi\/}\nolimits\!\left(1-z\right),$ $|z|<1$. 25.8.6 $\displaystyle\sum_{k=0}^{\infty}\mathop{\zeta\/}\nolimits\!\left(2k\right)z^{2k}$ $\displaystyle=-\tfrac{1}{2}\pi z\mathop{\cot\/}\nolimits\!\left(\pi z\right),$ $|z|<1$. 25.8.7 $\displaystyle\sum_{k=2}^{\infty}\frac{\mathop{\zeta\/}\nolimits\!\left(k\right% )}{k}z^{k}$ $\displaystyle=-\gamma z+\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!% \left(1-z\right),$ $|z|<1$. 25.8.8 $\displaystyle\sum_{k=1}^{\infty}\frac{\mathop{\zeta\/}\nolimits\!\left(2k% \right)}{k}z^{2k}$ $\displaystyle=\mathop{\ln\/}\nolimits\!\left(\frac{\pi z}{\mathop{\sin\/}% \nolimits\!\left(\pi z\right)}\right),$ $|z|<1$.
 25.8.9 $\sum_{k=1}^{\infty}\frac{\mathop{\zeta\/}\nolimits\!\left(2k\right)}{(2k+1)2^{% 2k}}=\frac{1}{2}-\frac{1}{2}\mathop{\ln\/}\nolimits 2.$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $k$: nonnegative integer Referenced by: §25.8 Permalink: http://dlmf.nist.gov/25.8.E9 Encodings: TeX, pMML, png See also: Annotations for 25.8
 25.8.10 $\sum_{k=1}^{\infty}\frac{\mathop{\zeta\/}\nolimits\!\left(2k\right)}{(2k+1)(2k% +2)2^{2k}}=\frac{1}{4}-\frac{7}{4\pi^{2}}\mathop{\zeta\/}\nolimits\!\left(3% \right).$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $k$: nonnegative integer Referenced by: §25.8 Permalink: http://dlmf.nist.gov/25.8.E10 Encodings: TeX, pMML, png See also: Annotations for 25.8

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).