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DLMF: §25.6 Integer Arguments ‣ Riemann Zeta Function ‣ Chapter 25 Zeta and Related Functions

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25 Zeta and Related Functions Riemann Zeta Function 25.5 Integral Representations 25.7 Integrals

§25.6 Integer Arguments

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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

§25.6(i) Function Values

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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		$\displaystyle\zeta\left(0\right)$	$\displaystyle=-\frac{1}{2},$
		$\displaystyle\zeta\left(2\right)$	$\displaystyle=\frac{\pi^{2}}{6},$
		$\displaystyle\zeta\left(4\right)$	$\displaystyle=\frac{\pi^{4}}{90},$
		$\displaystyle\zeta\left(6\right)$	$\displaystyle=\frac{\pi^{6}}{945}.$
ⓘ 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

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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		$\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi\right).$
ⓘ 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		$\displaystyle(-1)^{k}{\zeta}^{(k)}\left(-2n\right)$	$\displaystyle=\frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}% \genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Im\left(c^{k-m}% \right)\*{\Gamma}^{(r)}\left(2n+1\right){\zeta}^{(m-r)}\left(2n+1\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Im$ : 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		$\displaystyle(-1)^{k}{\zeta}^{(k)}\left(1-2n\right)$	$\displaystyle=\frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}% \genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Re\left(c^{k-m}% \right)\*{\Gamma}^{(r)}\left(2n\right){\zeta}^{(m-r)}\left(2n\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Re$ : 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		$\displaystyle\zeta'\left(2n\right)$	$\displaystyle=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\zeta'\left(1-2n% \right)-(\psi\left(2n\right)-\ln\left(2\pi\right))B_{2n}\right).$
ⓘ 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, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : 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		$\left(n+\tfrac{1}{2}\right)\zeta\left(2n\right)=\sum_{k=1}^{n-1}\zeta\left(2k% \right)\zeta\left(2n-2k\right),$
$n\geq 2$ .
ⓘ 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).

25.5 Integral Representations 25.7 Integrals

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