25 Zeta and Related FunctionsRiemann Zeta Function25.5 Integral Representations25.7 Integrals

§25.6 Integer Arguments

ⓘ

Notes:

See van der Poorten (1980, pp. 269–275), Apostol (1985a), and Basu and Apostol (2000).

Referenced by:

§24.4(ix)

Permalink:

http://dlmf.nist.gov/25.6

See also:

Annotations for Ch.25

Contents

1. §25.6(i) Function Values
2. §25.6(ii) Derivative Values
3. §25.6(iii) Recursion Formulas

§25.6(i) Function Values

ⓘ

Keywords:

Bernoulli and Euler numbers and polynomials, Riemann zeta function, integer argument, relations to other functions

Notes:

See Magnus et al. (1966, pp.19–20), Apostol (1976, p. 266), and van der Poorten (1980, pp. 269–275).

Referenced by:

§5.15

Permalink:

http://dlmf.nist.gov/25.6.i

See also:

Annotations for §25.6 and Ch.25

25.6.1		$\zeta \left(0\right)$	$=-{\displaystyle \frac{1}{2}},$
		$\zeta \left(2\right)$	$={\displaystyle \frac{{\pi }^2}{6}},$
		$\zeta \left(4\right)$	$={\displaystyle \frac{{\pi }^4}{90}},$
		$\zeta \left(6\right)$	$={\displaystyle \frac{{\pi }^6}{945}}.$
ⓘ Symbols: $\zeta \left(s\right)$: 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		$\zeta \left(2n\right)$	$={\displaystyle \frac{{(2\pi )}^{2n}}{2(2n)!}}\left|B_{2n}\right|,$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta \left(s\right)$: 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		$\zeta \left(-n\right)$	$=-{\displaystyle \frac{B_{n+1}}{n+1}},$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer Source: Apostol (1976, (20), p. 266); with $n\ne 0$ 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		$\zeta \left(-2n\right)$	$=0,$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta \left(s\right)$: 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		$$\zeta \left(k+1\right)=\frac{1}{k!}\sum _{n_1=1}^{\mathrm{\infty }}\mathrm{\dots }\sum _{n_k=1}^{\mathrm{\infty }}\frac{1}{n_1\mathrm{\cdots }{n}_k(n_1+\mathrm{\cdots }+n_k)},$$
$k=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta \left(s\right)$: 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		$$\zeta \left(2k+1\right)=\frac{{(-1)}^{k+1}{(2\pi )}^{2k+1}}{2(2k+1)!}{\int }_0^1{B}_{2k+1}\left(t\right)\cot \left(\pi t\right)dt,$$
$k=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: 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		$\zeta \left(2\right)$	$={\displaystyle {\int }_0^1}{\displaystyle {\int }_0^1}{\displaystyle \frac{1}{1-xy}}dxdy.$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $dx$: 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		$\zeta \left(2\right)$	$=3{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{k^2\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}}.$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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		$\zeta \left(3\right)$	$={\displaystyle \frac{5}{2}}{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{{(-1)}^{k-1}}{k^3\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}}.$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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		$\zeta \left(4\right)$	$={\displaystyle \frac{36}{17}}{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{k^4\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}}.$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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

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Keywords:

Riemann zeta function, derivatives, integer arguments

Notes:

See Apostol (1985a).

Permalink:

http://dlmf.nist.gov/25.6.ii

See also:

Annotations for §25.6 and Ch.25

25.6.11		$${\zeta }^{\prime }\left(0\right)=-\frac{1}{2}\ln \left(2\pi \right).$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: 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		$${\zeta }^{\prime \prime }\left(0\right)=-\frac{1}{2}{(\ln \left(2\pi \right))}^2+\frac{1}{2}{\gamma }^2-\frac{1}{24}{\pi }^2+{\gamma }_1,$$
ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: 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 $n=1,2,3,\mathrm{\dots }$,

25.6.13		${(-1)}^k{\zeta }^{(k)}\left(-2n\right)$	$={\displaystyle \frac{2{(-1)}^n}{{(2\pi )}^{2n+1}}}{\displaystyle \sum _{m=0}^k}{\displaystyle \sum _{r=0}^m}\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{m}}\right)\left({\displaystyle \genfrac{}{}{0.0pt}{}{m}{r}}\right)\mathrm{\Im }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n+1\right){\zeta }^{(m-r)}\left(2n+1\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$ Source: Apostol (1985a, p. 225) Permalink: http://dlmf.nist.gov/25.6.E13 Encodings: TeX, pMML, png See also: Annotations for §25.6(ii), §25.6 and Ch.25
25.6.14		${(-1)}^k{\zeta }^{(k)}\left(1-2n\right)$	$={\displaystyle \frac{2{(-1)}^n}{{(2\pi )}^{2n}}}{\displaystyle \sum _{m=0}^k}{\displaystyle \sum _{r=0}^m}\left({\displaystyle \genfrac{}{}{0.0pt}{}{k}{m}}\right)\left({\displaystyle \genfrac{}{}{0.0pt}{}{m}{r}}\right)\mathrm{\Re }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n\right){\zeta }^{(m-r)}\left(2n\right),$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$ Source: Apostol (1985a, p. 225) Permalink: http://dlmf.nist.gov/25.6.E14 Encodings: TeX, pMML, png See also: Annotations for §25.6(ii), §25.6 and Ch.25
25.6.15		${\zeta }^{\prime }\left(2n\right)$	$={\displaystyle \frac{{(-1)}^{n+1}{(2\pi )}^{2n}}{2(2n)!}}\left(2n{\zeta }^{\prime }\left(1-2n\right)-(\psi \left(2n\right)-\ln \left(2\pi \right))B_{2n}\right).$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function and $n$: nonnegative integer Source: Miller and Adamchik (1998, (2), p. 202) Referenced by: (25.11.21) Permalink: http://dlmf.nist.gov/25.6.E15 Encodings: TeX, pMML, png See also: Annotations for §25.6(ii), §25.6 and Ch.25

§25.6(iii) Recursion Formulas

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Keywords:

Riemann zeta function, recursion formulas

Notes:

See Basu and Apostol (2000, pp. 397, 404–406).

Permalink:

http://dlmf.nist.gov/25.6.iii

See also:

Annotations for §25.6 and Ch.25

25.6.16		$$\left(n+\frac{1}{2}\right)\zeta \left(2n\right)=\sum _{k=1}^{n-1}\zeta \left(2k\right)\zeta \left(2n-2k\right),$$
$n\ge 2$.
ⓘ Symbols: $\zeta \left(s\right)$: 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		$$\left(n+\frac{3}{4}\right)\zeta \left(4n+2\right)=\sum _{k=1}^n\zeta \left(2k\right)\zeta \left(4n+2-2k\right),$$
$n\ge 1$.
ⓘ Symbols: $\zeta \left(s\right)$: 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		$$\left(n+\frac{1}{4}\right)\zeta \left(4n\right)+\frac{1}{2}{(\zeta \left(2n\right))}^2=\sum _{k=1}^n\zeta \left(2k\right)\zeta \left(4n-2k\right),$$
$n\ge 1$.
ⓘ Symbols: $\zeta \left(s\right)$: 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		$$\left(m+n+\frac{3}{2}\right)\zeta \left(2m+2n+2\right)=\left(\sum _{k=1}^m+\sum _{k=1}^n\right)\zeta \left(2k\right)\zeta \left(2m+2n+2-2k\right),$$
$m\ge 0$, $n\ge 0$, $m+n\ge 1$.
ⓘ Symbols: $\zeta \left(s\right)$: 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		$$\frac{1}{2}(2^{2n}-1)\zeta \left(2n\right)=\sum _{k=1}^{n-1}(2^{2n-2k}-1)\zeta \left(2n-2k\right)\zeta \left(2k\right),$$
$n\ge 2$.
ⓘ Symbols: $\zeta \left(s\right)$: 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).

25.5 Integral Representations25.7 Integrals

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