§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

§25.6(i) Function Values
§25.6(ii) Derivative Values
§25.6(iii) Recursion Formulas

§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

$\zeta\left(0\right)=-\frac{1}{2},$

$\zeta\left(2\right)=\frac{\pi^{2}}{6},$

$\zeta\left(4\right)=\frac{\pi^{4}}{90},$

$\zeta\left(6\right)=\frac{\pi^{6}}{945}.$

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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 $\zeta\left(2n\right)=\frac{(2\pi)^{2n}}{2(2n)!}\left|B_{2n}\right|,$ $n=1,2,3,\dots$ .

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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 ) and : 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 $\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1},$ $n=1,2,3,\dots$ .

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : 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 $\zeta\left(-2n\right)=0,$ $n=1,2,3,\dots$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : 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 $\zeta\left(k+1\right)=\frac{1}{k!}\sum_{n_{1}=1}^{\infty}\dots\sum_{n_{k}=1}^{% \infty}\frac{1}{n_{1}\cdots n_{k}(n_{1}+\dots+n_{k})},$ $k=1,2,3,\dots$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : factorial (as in ), : nonnegative integer and : 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 $\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}B_{% 2k+1}\left(t\right)\cot\left(\pi t\right)\,\mathrm{d}t,$ $k=1,2,3,\dots$ .

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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 , : factorial (as in ), $\int$ : integral and : 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 $\zeta\left(2\right)=\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\,\mathrm{d}x\,% \mathrm{d}y.$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : 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 $\zeta\left(2\right)=3\sum_{k=1}^{\infty}\frac{1}{k^{2}\genfrac{(}{)}{0.0pt}{}{% 2k}{k}}.$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and : 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 $\zeta\left(3\right)=\frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}% \genfrac{(}{)}{0.0pt}{}{2k}{k}}.$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and : 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 $\zeta\left(4\right)=\frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\genfrac{(}{% )}{0.0pt}{}{2k}{k}}.$

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and : 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 $\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi\right).$

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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 $\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{% \gamma}^{2}-\tfrac{1}{24}\pi^{2}+\gamma_{1},$

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

25.6.13 $(-1)^{k}{\zeta}^{(k)}\left(-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0% }^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}% \Im\left(c^{k-m}\right)\*{\Gamma}^{(r)}\left(2n+1\right){\zeta}^{(m-r)}\left(2% n+1\right),$

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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, : nonnegative integer, : nonnegative integer, : nonnegative integer and
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 $(-1)^{k}{\zeta}^{(k)}\left(1-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}% ^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}% \Re\left(c^{k-m}\right)\*{\Gamma}^{(r)}\left(2n\right){\zeta}^{(m-r)}\left(2n% \right),$

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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, : nonnegative integer, : nonnegative integer, : nonnegative integer and
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 $\zeta'\left(2n\right)=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\zeta'\left(% 1-2n\right)-(\psi\left(2n\right)-\ln\left(2\pi\right))B_{2n}\right).$

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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 ), $\ln\NVar{z}$ : principal branch of logarithm function and : 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 $\left(n+\tfrac{1}{2}\right)\zeta\left(2n\right)=\sum_{k=1}^{n-1}\zeta\left(2k% \right)\zeta\left(2n-2k\right),$ $n\geq 2$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : nonnegative integer and : 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 $\left(n+\tfrac{3}{4}\right)\zeta\left(4n+2\right)=\sum_{k=1}^{n}\zeta\left(2k% \right)\zeta\left(4n+2-2k\right),$ $n\geq 1$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : nonnegative integer and : 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 ${\left(n+\tfrac{1}{4}\right)\zeta\left(4n\right)+\tfrac{1}{2}(\zeta\left(2n% \right))^{2}=\sum_{k=1}^{n}\zeta\left(2k\right)\zeta\left(4n-2k\right)},$ $n\geq 1$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : nonnegative integer and : 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 $\left(m+n+\tfrac{3}{2}\right)\zeta\left(2m+2n+2\right)=\left(\sum_{k=1}^{m}+% \sum_{k=1}^{n}\right)\zeta\left(2k\right)\zeta\left(2m+2n+2-2k\right),$ $m\geq 0$ , $n\geq 0$ , $m+n\geq 1$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : nonnegative integer, : nonnegative integer and : 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 $\tfrac{1}{2}(2^{2n}-1)\zeta\left(2n\right)=\sum_{k=1}^{n-1}(2^{2n-2k}-1)\zeta% \left(2n-2k\right)\zeta\left(2k\right),$ $n\geq 2$ .

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, : nonnegative integer and : 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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§25.6 Integer Arguments

Contents

§25.6(i) Function Values

§25.6(ii) Derivative Values

§25.6(iii) Recursion Formulas