§25.8 Sums

Notes:: See Adamchik and Srivastava (1998) and Erdélyi et al. (1953a, pp. 45 and 51). For (25.8.2) see Landau (1953, p. 274). For (25.8.3) see Srivastava (1988). For (25.8.7), (25.8.8) divide by $x$ in (25.8.5), (25.8.6) and integrate. For (25.8.9) see Srivastava and Choi (2001, p. 212). For (25.8.10) see Ewell (1990).
Keywords:: Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.8

25.8.1 $\sum _{{k=2}}^{\infty}\left(\mathop{\zeta\/}\nolimits\!\left(k\right)-1\right)=1.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.8.E1
Encodings:: TeX, pMML, png

25.8.2 $\sum _{{k=0}}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+k\right)}{(k+1)!}\left(\mathop{\zeta\/}\nolimits\!\left(s+k\right)-1\right)=\mathop{\Gamma\/}\nolimits\!\left(s-1\right),$ $s\neq 1,0,-1,-2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $!$ : $n!$ : factorial, $k$ : nonnegative integer and $s$ : complex variable
Referenced by:: §25.8
Permalink:: http://dlmf.nist.gov/25.8.E2
Encodings:: TeX, pMML, png

25.8.3 $\sum _{{k=0}}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+k\right)\mathop{\zeta\/}\nolimits\!\left(s+k\right)}{k!\mathop{\Gamma\/}\nolimits\!\left(s\right)2^{{s+k}}}=(1-2^{{-s}})\mathop{\zeta\/}\nolimits\!\left(s\right),$ $s\neq 1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $!$ : $n!$ : factorial, $k$ : nonnegative integer and $s$ : complex variable
Referenced by:: §25.11(iv), §25.8
Permalink:: http://dlmf.nist.gov/25.8.E3
Encodings:: TeX, pMML, png

25.8.4 $\sum _{{k=1}}^{\infty}\frac{(-1)^{k}}{k}(\mathop{\zeta\/}\nolimits\!\left(nk\right)-1)=\mathop{\ln\/}\nolimits\!\left(\prod _{{j=0}}^{{n-1}}\mathop{\Gamma\/}\nolimits\!\left(2-e^{{(2j+1)\pi i/n}}\right)\right),$ $n=2,3,4,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $e$ : base of exponential function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.8.E4
Encodings:: TeX, pMML, png

25.8.5 $\sum _{{k=2}}^{\infty}\mathop{\zeta\/}\nolimits\!\left(k\right)z^{k}=-\EulerConstant z-z\mathop{\psi\/}\nolimits\!\left(1-z\right),$ $|z|<1$ .

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §25.8
Permalink:: http://dlmf.nist.gov/25.8.E5
Encodings:: TeX, pMML, png

25.8.6 $\sum _{{k=0}}^{\infty}\mathop{\zeta\/}\nolimits\!\left(2k\right)z^{{2k}}=-\tfrac{1}{2}\pi z\mathop{\cot\/}\nolimits\!\left(\pi z\right),$ $|z|<1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §25.8
Permalink:: http://dlmf.nist.gov/25.8.E6
Encodings:: TeX, pMML, png

25.8.7 $\sum _{{k=2}}^{\infty}\frac{\mathop{\zeta\/}\nolimits\!\left(k\right)}{k}z^{k}=-\EulerConstant z+\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(1-z\right),$ $|z|<1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §25.8
Permalink:: http://dlmf.nist.gov/25.8.E7
Encodings:: TeX, pMML, png

25.8.8 $\sum _{{k=1}}^{\infty}\frac{\mathop{\zeta\/}\nolimits\!\left(2k\right)}{k}z^{{2k}}=\mathop{\ln\/}\nolimits\!\left(\frac{\pi z}{\mathop{\sin\/}\nolimits\!\left(\pi z\right)}\right),$ $|z|<1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §25.8
Permalink:: http://dlmf.nist.gov/25.8.E8
Encodings:: TeX, pMML, png

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:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : 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

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:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $k$ : nonnegative integer
Referenced by:: §25.8
Permalink:: http://dlmf.nist.gov/25.8.E10
Encodings:: TeX, pMML, png

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).