§25.6 Integer Arguments

Referenced by:: §24.4(ix)
Permalink:: http://dlmf.nist.gov/25.6

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

§25.6(i) Function Values

Notes:: For (25.6.1)–(25.6.4) see Apostol (1976, pp. 266–268). For (25.6.5) see Mordell (1958). For (25.6.6) see Nörlund (1924, p. 66). For (25.6.7) see Apostol (1983). For (25.6.8)–(25.6.10) see van der Poorten (1980, pp. 271 and 274).
Keywords:: Riemann zeta function
Referenced by:: §5.15
Permalink:: http://dlmf.nist.gov/25.6.i

25.6.1

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

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

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

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

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function
A&S Ref:: 23.2.11
Referenced by:: §25.6(i)
Permalink:: http://dlmf.nist.gov/25.6.E1
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

25.6.2 $\mathop{\zeta\/}\nolimits\!\left(2n\right)=\frac{(2\pi)^{{2n}}}{2(2n)!}\left|\mathop{B_{{2n}}\/}\nolimits\right|,$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $!$ : $n!$ : factorial and $n$ : nonnegative integer
A&S Ref:: 23.2.16
Permalink:: http://dlmf.nist.gov/25.6.E2
Encodings:: TeX, pMML, png

25.6.3 $\mathop{\zeta\/}\nolimits\!\left(-n\right)=-\frac{\mathop{B_{{n+1}}\/}\nolimits}{n+1},$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $n$ : nonnegative integer
A&S Ref:: 23.2.15 (in slightly different form)
Permalink:: http://dlmf.nist.gov/25.6.E3
Encodings:: TeX, pMML, png

25.6.4 $\mathop{\zeta\/}\nolimits\!\left(-2n\right)=0,$ $n=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $n$ : nonnegative integer
A&S Ref:: 23.2.14
Referenced by:: §25.6(i)
Permalink:: http://dlmf.nist.gov/25.6.E4
Encodings:: TeX, pMML, png

25.6.5 $\mathop{\zeta\/}\nolimits\!\left(k+1\right)=\frac{1}{k!}\sum _{{n_{1}=1}}^{\infty}\dots\sum _{{n_{k}=1}}^{\infty}\frac{1}{n_{1}\cdots n_{k}(n_{1}+\dots+n_{k})},$ $k=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $!$ : $n!$ : factorial, $k$ : nonnegative integer and $n$ : nonnegative integer
Referenced by:: §25.6(i)
Permalink:: http://dlmf.nist.gov/25.6.E5
Encodings:: TeX, pMML, png

25.6.6 $\mathop{\zeta\/}\nolimits\!\left(2k+1\right)=\frac{(-1)^{{k+1}}(2\pi)^{{2k+1}}}{2(2k+1)!}\int _{0}^{1}\mathop{B_{{2k+1}}\/}\nolimits\!\left(t\right)\mathop{\cot\/}\nolimits\!\left(\pi t\right)dt,$ $k=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\cot\/}\nolimits z$ : cotangent function, $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral and $k$ : nonnegative integer
A&S Ref:: 23.2.17
Referenced by:: §25.6(i)
Permalink:: http://dlmf.nist.gov/25.6.E6
Encodings:: TeX, pMML, png

25.6.7 $\mathop{\zeta\/}\nolimits\!\left(2\right)=\int _{0}^{1}\int _{0}^{1}\frac{1}{1-xy}dxdy.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\int$ : integral and $x$ : real variable
Referenced by:: §25.6(i)
Permalink:: http://dlmf.nist.gov/25.6.E7
Encodings:: TeX, pMML, png

25.6.8 $\mathop{\zeta\/}\nolimits\!\left(2\right)=3\sum _{{k=1}}^{\infty}\frac{1}{k^{2}\binom{2k}{k}}.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient and $k$ : nonnegative integer
Referenced by:: §25.6(i)
Permalink:: http://dlmf.nist.gov/25.6.E8
Encodings:: TeX, pMML, png

25.6.9 $\mathop{\zeta\/}\nolimits\!\left(3\right)=\frac{5}{2}\sum _{{k=1}}^{\infty}\frac{(-1)^{{k-1}}}{k^{3}\binom{2k}{k}}.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.6.E9
Encodings:: TeX, pMML, png

25.6.10 $\mathop{\zeta\/}\nolimits\!\left(4\right)=\frac{36}{17}\sum _{{k=1}}^{\infty}\frac{1}{k^{4}\binom{2k}{k}}.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient and $k$ : nonnegative integer
Referenced by:: §25.6(i)
Permalink:: http://dlmf.nist.gov/25.6.E10
Encodings:: TeX, pMML, png

§25.6(ii) Derivative Values

Notes:: See Apostol (1985a). For (25.6.15) see Miller and Adamchik (1998).
Keywords:: Riemann zeta function, derivatives
Permalink:: http://dlmf.nist.gov/25.6.ii

25.6.11 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(0\right)=-\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(2\pi\right).$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
A&S Ref:: 23.2.13
Permalink:: http://dlmf.nist.gov/25.6.E11
Encodings:: TeX, pMML, png

25.6.12 ${\mathop{\zeta\/}\nolimits^{{\prime\prime}}}\!\left(0\right)=-\tfrac{1}{2}(\mathop{\ln\/}\nolimits\!\left(2\pi\right))^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\gamma _{1},$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $\gamma _{n}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/25.6.E12
Encodings:: TeX, pMML, png

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 $(-1)^{k}{\mathop{\zeta\/}\nolimits^{{(k)}}}\!\left(-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{{2n+1}}}\sum _{{m=0}}^{k}\sum _{{r=0}}^{m}\binom{k}{m}\binom{m}{r}\imagpart{(c^{{k-m}})}\*{\mathop{\Gamma\/}\nolimits^{{(r)}}}\!\left(2n+1\right){\mathop{\zeta\/}\nolimits^{{(m-r)}}}\!\left(2n+1\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $\imagpart{}$ : imaginary part, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $c$
Permalink:: http://dlmf.nist.gov/25.6.E13
Encodings:: TeX, pMML, png

25.6.14 $(-1)^{k}{\mathop{\zeta\/}\nolimits^{{(k)}}}\!\left(1-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{{2n}}}\sum _{{m=0}}^{k}\sum _{{r=0}}^{m}\binom{k}{m}\binom{m}{r}\realpart{(c^{{k-m}})}\*{\mathop{\Gamma\/}\nolimits^{{(r)}}}\!\left(2n\right){\mathop{\zeta\/}\nolimits^{{(m-r)}}}\!\left(2n\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $\realpart{}$ : real part, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $c$
Permalink:: http://dlmf.nist.gov/25.6.E14
Encodings:: TeX, pMML, png

25.6.15 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(2n\right)=\frac{(-1)^{{n+1}}(2\pi)^{{2n}}}{2(2n)!}\left(2n{\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(1-2n\right)-(\mathop{\psi\/}\nolimits\!\left(2n\right)-\mathop{\ln\/}\nolimits\!\left(2\pi\right))\mathop{B_{{2n}}\/}\nolimits\right).$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $n$ : nonnegative integer
Referenced by:: §25.6(ii)
Permalink:: http://dlmf.nist.gov/25.6.E15
Encodings:: TeX, pMML, png

§25.6(iii) Recursion Formulas

Notes:: See Basu and Apostol (2000).
Keywords:: Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.6.iii

25.6.16 $\left(n+\tfrac{1}{2}\right)\mathop{\zeta\/}\nolimits\!\left(2n\right)=\sum _{{k=1}}^{{n-1}}\mathop{\zeta\/}\nolimits\!\left(2k\right)\mathop{\zeta\/}\nolimits\!\left(2n-2k\right),$ $n\geq 2$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.6.E16
Encodings:: TeX, pMML, png

25.6.17 $\left(n+\tfrac{3}{4}\right)\mathop{\zeta\/}\nolimits\!\left(4n+2\right)=\sum _{{k=1}}^{n}\mathop{\zeta\/}\nolimits\!\left(2k\right)\mathop{\zeta\/}\nolimits\!\left(4n+2-2k\right),$ $n\geq 1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.6.E17
Encodings:: TeX, pMML, png

25.6.18 ${\left(n+\tfrac{1}{4}\right)\mathop{\zeta\/}\nolimits\!\left(4n\right)+\tfrac{1}{2}(\mathop{\zeta\/}\nolimits\!\left(2n\right))^{2}=\sum _{{k=1}}^{n}\mathop{\zeta\/}\nolimits\!\left(2k\right)\mathop{\zeta\/}\nolimits\!\left(4n-2k\right)},$ $n\geq 1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.6.E18
Encodings:: TeX, pMML, png

25.6.19 $\left(m+n+\tfrac{3}{2}\right)\mathop{\zeta\/}\nolimits\!\left(2m+2n+2\right)=\left(\sum _{{k=1}}^{m}+\sum _{{k=1}}^{n}\right)\mathop{\zeta\/}\nolimits\!\left(2k\right)\mathop{\zeta\/}\nolimits\!\left(2m+2n+2-2k\right),$ $m\geq 0$ , $n\geq 0$ , $m+n\geq 1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.6.E19
Encodings:: TeX, pMML, png

25.6.20 $\tfrac{1}{2}(2^{{2n}}-1)\mathop{\zeta\/}\nolimits\!\left(2n\right)=\sum _{{k=1}}^{{n-1}}(2^{{2n-2k}}-1)\mathop{\zeta\/}\nolimits\!\left(2n-2k\right)\mathop{\zeta\/}\nolimits\!\left(2k\right),$ $n\geq 2$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $k$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/25.6.E20
Encodings:: TeX, pMML, png

For related results see Basu and Apostol (2000).

§25.6 Integer Arguments

Contents

§25.6(i) Function Values

§25.6(ii) Derivative Values

§25.6(iii) Recursion Formulas