§25.6 Integer Arguments

§25.6(i) Function Values

 25.6.1 $\displaystyle\zeta\left(0\right)$ $\displaystyle=-\frac{1}{2},$ $\displaystyle\zeta\left(2\right)$ $\displaystyle=\frac{\pi^{2}}{6},$ $\displaystyle\zeta\left(4\right)$ $\displaystyle=\frac{\pi^{4}}{90},$ $\displaystyle\zeta\left(6\right)$ $\displaystyle=\frac{\pi^{6}}{945}.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $\pi$: the ratio of the circumference of a circle to its diameter Source: Magnus et al. (1966, p. 19); with Table 24.2.1 A&S Ref: 23.2.11 Referenced by: (25.16.8) Permalink: http://dlmf.nist.gov/25.6.E1 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §25.6(i), §25.6 and Ch.25
 25.6.2 $\displaystyle\zeta\left(2n\right)$ $\displaystyle=\frac{(2\pi)^{2n}}{2(2n)!}\left|B_{2n}\right|,$ $n=1,2,3,\dots$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: nonnegative integer Sources: Magnus et al. (1966, p. 19); Apostol (1976, (22), p. 266) A&S Ref: 23.2.16 Referenced by: (25.11.21) Permalink: http://dlmf.nist.gov/25.6.E2 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25 25.6.3 $\displaystyle\zeta\left(-n\right)$ $\displaystyle=-\frac{B_{n+1}}{n+1},$ $n=1,2,3,\dots$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $n$: nonnegative integer Source: Apostol (1976, (20), p. 266); with $n\neq 0$ A&S Ref: 23.2.15 (in slightly different form) Permalink: http://dlmf.nist.gov/25.6.E3 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25 25.6.4 $\displaystyle\zeta\left(-2n\right)$ $\displaystyle=0,$ $n=1,2,3,\dots$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $n$: nonnegative integer Source: Apostol (1976, Theorem 12.16, p. 266) A&S Ref: 23.2.14 Permalink: http://dlmf.nist.gov/25.6.E4 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25
 25.6.5 $\zeta\left(k+1\right)=\frac{1}{k!}\sum_{n_{1}=1}^{\infty}\dots\sum_{n_{k}=1}^{% \infty}\frac{1}{n_{1}\cdots n_{k}(n_{1}+\dots+n_{k})},$ $k=1,2,3,\dots$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $n$: nonnegative integer Source: Mordell (1958, (5), p. 369) Permalink: http://dlmf.nist.gov/25.6.E5 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25
 25.6.6 $\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}B_{% 2k+1}\left(t\right)\cot\left(\pi t\right)\,\mathrm{d}t,$ $k=1,2,3,\dots$. ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$: Bernoulli polynomials, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$: cotangent function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral and $k$: nonnegative integer Source: Nörlund (1924, (81*), p. 66); with $\nu=k$ A&S Ref: 23.2.17 Permalink: http://dlmf.nist.gov/25.6.E6 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25
 25.6.7 $\displaystyle\zeta\left(2\right)$ $\displaystyle=\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\,\mathrm{d}x\,\mathrm{d}y.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $x$: real variable Keywords: double integral Source: Apostol (1983, p. 59) Permalink: http://dlmf.nist.gov/25.6.E7 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25 25.6.8 $\displaystyle\zeta\left(2\right)$ $\displaystyle=3\sum_{k=1}^{\infty}\frac{1}{k^{2}\genfrac{(}{)}{0.0pt}{}{2k}{k}}.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient and $k$: nonnegative integer Keywords: infinite series Source: van der Poorten (1980, (2), p. 271) Permalink: http://dlmf.nist.gov/25.6.E8 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25 25.6.9 $\displaystyle\zeta\left(3\right)$ $\displaystyle=\frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\genfrac{(}% {)}{0.0pt}{}{2k}{k}}.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient and $k$: nonnegative integer Keywords: infinite series Source: van der Poorten (1980, (3), p. 271) Permalink: http://dlmf.nist.gov/25.6.E9 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25 25.6.10 $\displaystyle\zeta\left(4\right)$ $\displaystyle=\frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\genfrac{(}{)}{0.0% pt}{}{2k}{k}}.$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient and $k$: nonnegative integer Keywords: infinite series Source: van der Poorten (1980, (5), p. 274) Permalink: http://dlmf.nist.gov/25.6.E10 Encodings: TeX, pMML, png See also: Annotations for §25.6(i), §25.6 and Ch.25

§25.6(ii) Derivative Values

 25.6.11 $\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi\right).$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$: principal branch of logarithm function Source: Apostol (1985a, p. 223) A&S Ref: 23.2.13 Permalink: http://dlmf.nist.gov/25.6.E11 Encodings: TeX, pMML, png See also: Annotations for §25.6(ii), §25.6 and Ch.25
 25.6.12 $\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{% \gamma}^{2}-\tfrac{1}{24}\pi^{2}+\gamma_{1},$ ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\gamma_{\NVar{n}}$: Stieltjes constants, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln\NVar{z}$: principal branch of logarithm function Source: Apostol (1985a, pp. 226, 227, 229) Referenced by: Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series Permalink: http://dlmf.nist.gov/25.6.E12 Encodings: TeX, pMML, png See also: Annotations for §25.6(ii), §25.6 and Ch.25

where $\gamma_{1}$ is given by (25.2.5).

With $c$ defined by (25.4.6) and $n=1,2,3,\dots$,

 25.6.13 $\displaystyle(-1)^{k}{\zeta}^{(k)}\left(-2n\right)$ $\displaystyle=\frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}% \genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Im\left(c^{k-m}% \right)\*{\Gamma}^{(r)}\left(2n+1\right){\zeta}^{(m-r)}\left(2n+1\right),$ 25.6.14 $\displaystyle(-1)^{k}{\zeta}^{(k)}\left(1-2n\right)$ $\displaystyle=\frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}% \genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Re\left(c^{k-m}% \right)\*{\Gamma}^{(r)}\left(2n\right){\zeta}^{(m-r)}\left(2n\right),$ 25.6.15 $\displaystyle\zeta'\left(2n\right)$ $\displaystyle=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\zeta'\left(1-2n% \right)-(\psi\left(2n\right)-\ln\left(2\pi\right))B_{2n}\right).$

§25.6(iii) Recursion Formulas

 25.6.16 $\left(n+\tfrac{1}{2}\right)\zeta\left(2n\right)=\sum_{k=1}^{n-1}\zeta\left(2k% \right)\zeta\left(2n-2k\right),$ $n\geq 2$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Source: Basu and Apostol (2000, (1.5), p. 397) Permalink: http://dlmf.nist.gov/25.6.E16 Encodings: TeX, pMML, png See also: Annotations for §25.6(iii), §25.6 and Ch.25
 25.6.17 $\left(n+\tfrac{3}{4}\right)\zeta\left(4n+2\right)=\sum_{k=1}^{n}\zeta\left(2k% \right)\zeta\left(4n+2-2k\right),$ $n\geq 1$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Source: Basu and Apostol (2000, (3.14), p. 404) Permalink: http://dlmf.nist.gov/25.6.E17 Encodings: TeX, pMML, png See also: Annotations for §25.6(iii), §25.6 and Ch.25
 25.6.18 ${\left(n+\tfrac{1}{4}\right)\zeta\left(4n\right)+\tfrac{1}{2}(\zeta\left(2n% \right))^{2}=\sum_{k=1}^{n}\zeta\left(2k\right)\zeta\left(4n-2k\right)},$ $n\geq 1$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Source: Basu and Apostol (2000, (3.15), p. 405) Permalink: http://dlmf.nist.gov/25.6.E18 Encodings: TeX, pMML, png See also: Annotations for §25.6(iii), §25.6 and Ch.25
 25.6.19 $\left(m+n+\tfrac{3}{2}\right)\zeta\left(2m+2n+2\right)=\left(\sum_{k=1}^{m}+% \sum_{k=1}^{n}\right)\zeta\left(2k\right)\zeta\left(2m+2n+2-2k\right),$ $m\geq 0$, $n\geq 0$, $m+n\geq 1$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer Source: Basu and Apostol (2000, (3.12), p. 404) Permalink: http://dlmf.nist.gov/25.6.E19 Encodings: TeX, pMML, png See also: Annotations for §25.6(iii), §25.6 and Ch.25
 25.6.20 $\tfrac{1}{2}(2^{2n}-1)\zeta\left(2n\right)=\sum_{k=1}^{n-1}(2^{2n-2k}-1)\zeta% \left(2n-2k\right)\zeta\left(2k\right),$ $n\geq 2$. ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $k$: nonnegative integer and $n$: nonnegative integer Source: Basu and Apostol (2000, (3.18), p. 406) Permalink: http://dlmf.nist.gov/25.6.E20 Encodings: TeX, pMML, png See also: Annotations for §25.6(iii), §25.6 and Ch.25

For related results see Basu and Apostol (2000).