§25.2 Definition and Expansions

Keywords:: Riemann zeta function
Referenced by:: §20.10(i), §20.9(iii)
Permalink:: http://dlmf.nist.gov/25.2

§25.2(i) Definition
§25.2(ii) Other Infinite Series
§25.2(iii) Representations by the Euler–Maclaurin Formula
§25.2(iv) Infinite Products

§25.2(i) Definition

Notes:: See Apostol (1976, p. 255).
Keywords:: Riemann zeta function
Referenced by:: ¶ ‣ §2.10(i), §26.12(iv), §27.1, §27.18, §27.4, §8.22(ii)
Permalink:: http://dlmf.nist.gov/25.2.i

When $\realpart{s}>1$ ,

25.2.1 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum _{{n=1}}^{\infty}\frac{1}{n^{s}}.$

Defines:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function
Symbols:: $n$ : nonnegative integer and $s$ : complex variable
A&S Ref:: 23.2.1
Referenced by:: §25.2(ii)
Permalink:: http://dlmf.nist.gov/25.2.E1
Encodings:: TeX, pMML, png

Elsewhere $\mathop{\zeta\/}\nolimits\!\left(s\right)$ is defined by analytic continuation. It is a meromorphic function whose only singularity in $\Complex$ is a simple pole at $s=1$ , with residue 1.

§25.2(ii) Other Infinite Series

Notes:: See Apostol (1976, pp. 236 and 292) and Hardy (1912).
Keywords:: Dirichlet series, Riemann zeta function, derivatives
Permalink:: http://dlmf.nist.gov/25.2.ii

25.2.2 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{1-2^{{-s}}}\sum _{{n=0}}^{\infty}\frac{1}{(2n+1)^{s}},$ $\realpart{s}>1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : real part, $n$ : nonnegative integer and $s$ : complex variable
A&S Ref:: 23.2.20 (is the special case with integer values of $s$ )
Referenced by:: §25.11(v)
Permalink:: http://dlmf.nist.gov/25.2.E2
Encodings:: TeX, pMML, png

25.2.3 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{1-2^{{1-s}}}\sum _{{n=1}}^{\infty}\frac{(-1)^{{n-1}}}{n^{s}},$ $\realpart{s}>0$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : real part, $n$ : nonnegative integer and $s$ : complex variable
A&S Ref:: 23.2.19 (is the special case with integer values of $s$ )
Referenced by:: §25.11(x)
Permalink:: http://dlmf.nist.gov/25.2.E3
Encodings:: TeX, pMML, png

25.2.4 ${\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{s-1}+\sum _{{n=0}}^{\infty}\frac{(-1)^{n}}{n!}\gamma _{n}(s-1)^{n},}$ $\realpart{s}>0$ ,

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $!$ : $n!$ : factorial, $\realpart{}$ : real part, $n$ : nonnegative integer, $s$ : complex variable and $\gamma _{n}$ : expansion coefficient
A&S Ref:: 23.2.5
Permalink:: http://dlmf.nist.gov/25.2.E4
Encodings:: TeX, pMML, png

where

25.2.5 $\gamma _{n}=\lim _{{m\to\infty}}\left(\sum _{{k=1}}^{m}\frac{(\mathop{\ln\/}\nolimits k)^{n}}{k}-\frac{(\mathop{\ln\/}\nolimits m)^{{n+1}}}{n+1}\right).$

Symbols:: $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $\gamma _{n}$ : expansion coefficient
Referenced by:: §25.6(ii)
Permalink:: http://dlmf.nist.gov/25.2.E5
Encodings:: TeX, pMML, png

25.2.6 ${\mathop{\zeta\/}\nolimits^{{\prime}}}\!\left(s\right)=-\sum _{{n=2}}^{\infty}(\mathop{\ln\/}\nolimits n)n^{{-s}},$ $\realpart{s}>1$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part, $n$ : nonnegative integer and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.2.E6
Encodings:: TeX, pMML, png

25.2.7 ${\mathop{\zeta\/}\nolimits^{{(k)}}}\!\left(s\right)=(-1)^{k}\sum _{{n=2}}^{\infty}(\mathop{\ln\/}\nolimits n)^{k}n^{{-s}},$ $\realpart{s}>1$ , $k=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.2.E7
Encodings:: TeX, pMML, png

For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. This includes, for example, $1/\mathop{\zeta\/}\nolimits\!\left(s\right)$ .

§25.2(iii) Representations by the Euler–Maclaurin Formula

Notes:: See Apostol (1976, p. 269) and Knopp (1948, p. 533). (25.2.9) follows from (25.2.8) by repeated integration by parts.
Keywords:: Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.2.iii

25.2.8 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum _{{k=1}}^{N}\frac{1}{k^{s}}+\frac{N^{{1-s}}}{s-1}-s\int _{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{{s+1}}}dx,$ $\realpart{s}>0$ , $N=1,2,3,\dots$ .

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\left\lfloor x\right\rfloor$ : floor of $x$ , $\int$ : integral, $\realpart{}$ : real part, $k$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
A&S Ref:: 23.2.9
Referenced by:: §25.2(iii)
Permalink:: http://dlmf.nist.gov/25.2.E8
Encodings:: TeX, pMML, png

25.2.9 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\sum _{{k=1}}^{N}\frac{1}{k^{s}}+\frac{N^{{1-s}}}{s-1}-\frac{1}{2}N^{{-s}}+\sum _{{k=1}}^{n}\binom{s+2k-2}{2k-1}\frac{\mathop{B_{{2k}}\/}\nolimits}{2k}N^{{1-s-2k}}-\binom{s+2n}{2n+1}\int _{N}^{\infty}\frac{\mathop{\widetilde{B}_{{2n+1}}\/}\nolimits\!\left(x\right)}{x^{{s+2n+1}}}dx,$ $\realpart{s}>-2n$ ; $n,N=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.18(i), §25.2(iii)
Permalink:: http://dlmf.nist.gov/25.2.E9
Encodings:: TeX, pMML, png

25.2.10 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{{k=1}}^{n}\binom{s+2k-2}{2k-1}\frac{\mathop{B_{{2k}}\/}\nolimits}{2k}-\binom{s+2n}{2n+1}\int _{1}^{\infty}\frac{\mathop{\widetilde{B}_{{2n+1}}\/}\nolimits\!\left(x\right)}{x^{{s+2n+1}}}dx,$ $\realpart{s}>-2n$ , $n=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\binom{m}{n}$ : binomial coefficient, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ : periodic Bernoulli functions, $\realpart{}$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
A&S Ref:: 23.2.3
Permalink:: http://dlmf.nist.gov/25.2.E10
Encodings:: TeX, pMML, png

For $\mathop{B_{{2k}}\/}\nolimits$ see §24.2(i), and for $\mathop{\widetilde{B}_{{n}}\/}\nolimits\!\left(x\right)$ see §24.2(iii).

§25.2(iv) Infinite Products

Notes:: See Apostol (1976, p. 231) and Titchmarsh (1986b, p. 30).
Keywords:: Riemann zeta function, infinite products
Permalink:: http://dlmf.nist.gov/25.2.iv

25.2.11 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\prod _{p}(1-p^{{-s}})^{{-1}},$ $\realpart{s}>1$ ,

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\realpart{}$ : real part, $p$ : prime number and $s$ : complex variable
A&S Ref:: 23.2.2
Referenced by:: §25.10(i)
Permalink:: http://dlmf.nist.gov/25.2.E11
Encodings:: TeX, pMML, png

product over all primes $p$ .

25.2.12 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{(2\pi)^{s}e^{{-s-(\EulerConstant s/2)}}}{2(s-1)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}s+1\right)}\prod _{\rho}\left(1-\frac{s}{\rho}\right)e^{{s/\rho}},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $e$ : base of exponential function, $s$ : complex variable and $\rho$ : zeros
A&S Ref:: 23.2.10 (in slightly different form)
Permalink:: http://dlmf.nist.gov/25.2.E12
Encodings:: TeX, pMML, png

product over zeros $\rho$ of $\mathop{\zeta\/}\nolimits$ with $\realpart{\rho}>0$ (see §25.10(i)); $\EulerConstant$ is Euler’s constant (§5.2(ii)).

§25.2 Definition and Expansions

Contents

§25.2(i) Definition

§25.2(ii) Other Infinite Series

§25.2(iii) Representations by the Euler–Maclaurin Formula

§25.2(iv) Infinite Products