§25.2 Definition and Expansions

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Keywords:: Riemann zeta function, analytic properties
Notes:: See Apostol (1976, Chapter 12) and Ivić (1985, Section 1.1).
Referenced by:: §20.10(i), §20.9(iii)
Permalink:: http://dlmf.nist.gov/25.2
See also:: Annotations for Ch.25

§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

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Keywords:: Riemann zeta function, definition
Notes:: See Apostol (1976, Chapter 12), Ivić (1985, Section 1.1) and Riemann (1859, p. 136).
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
See also:: Annotations for §25.2 and Ch.25

When $\Re s>1$ ,

25.2.1		$\zeta\left(s\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$
ⓘ Defines: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function Symbols: $n$ : nonnegative integer and $s$ : complex variable Keywords: definition, infinite series Sources: Apostol (1976, Chapter 12); Riemann (1859, p. 136) A&S Ref: 23.2.1 Referenced by: (25.11.2), (25.2.2), (25.2.4), §25.2(ii), Erratum (V1.0.17) for Equation (25.2.4) Permalink: http://dlmf.nist.gov/25.2.E1 Encodings: TeX, pMML, png See also: Annotations for §25.2(i), §25.2 and Ch.25

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

§25.2(ii) Other Infinite Series

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Keywords:: Dirichlet series, Riemann zeta function, derivatives, series expansions
Notes:: See Hardy (1912, pp. 215–216) and Ivić (1985, Section 1.1).
Referenced by:: Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series
Permalink:: http://dlmf.nist.gov/25.2.ii
Addition (effective with 1.0.23):: Immediately above (25.2.5), it is now mentioned that $\gamma_{n}$ defines the Stieltjes constants.
See also:: Annotations for §25.2 and Ch.25

25.2.2		$\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}},$
$\Re s>1$ .
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable Keywords: infinite series Proof sketch: Derivable from (25.2.1). A&S Ref: 23.2.20 (is the special case with integer values of $s$ ) Referenced by: (25.11.11) Permalink: http://dlmf.nist.gov/25.2.E2 Encodings: TeX, pMML, png See also: Annotations for §25.2(ii), §25.2 and Ch.25

25.2.3		$\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^% {s}},$
$\Re s>0$ .
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable Keywords: infinite series Source: Apostol (1976, (27), p. 292) 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 See also: Annotations for §25.2(ii), §25.2 and Ch.25

25.2.4		$\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\gamma_{\NVar{n}}$ : Stieltjes constants, $!$ : factorial (as in $n!$ ), $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable Keywords: Laurent series, infinite series Sources: Hardy (1912, p. 216); mistakenly missing factor of $(-1)^{n}/n!$ ; Ivić (1985, (1.11), p. 4) A&S Ref: 23.2.5 (with unnecessary constraint removed) Referenced by: Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series Permalink: http://dlmf.nist.gov/25.2.E4 Encodings: TeX, pMML, png Clarification (effective with 1.0.17): Originally this equation had the constraint $\Re s>0$ . This constraint is unnecessary because, as stated after (25.2.1), $\zeta\left(s\right)$ is meromorphic with a simple pole at $s=1$ . Suggested 2017-12-05 by John Harper See also: Annotations for §25.2(ii), §25.2 and Ch.25

where the Stieltjes constants $\gamma_{n}$ are defined via

25.2.5		$\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(% \ln m)^{n+1}}{n+1}\right).$
ⓘ Defines: $\gamma_{\NVar{n}}$ : Stieltjes constants Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer Keywords: Stieltjes constants, definition Sources: Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4) Referenced by: §25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series Permalink: http://dlmf.nist.gov/25.2.E5 Encodings: TeX, pMML, png See also: Annotations for §25.2(ii), §25.2 and Ch.25

25.2.6		$\zeta'\left(s\right)=-\sum_{n=2}^{\infty}(\ln n)n^{-s},$
$\Re s>1$ .
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $n$ : nonnegative integer and $s$ : complex variable Keywords: infinite series Source: Apostol (1976, (12), p. 236) Permalink: http://dlmf.nist.gov/25.2.E6 Encodings: TeX, pMML, png See also: Annotations for §25.2(ii), §25.2 and Ch.25

25.2.7		${\zeta}^{(k)}\left(s\right)=(-1)^{k}\sum_{n=2}^{\infty}(\ln n)^{k}n^{-s},$
$\Re s>1$ , $k=1,2,3,\dots$ .
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\ln\NVar{z}$ : principal branch of logarithm function, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer and $s$ : complex variable Keywords: infinite series Source: Apostol (1976, p. 236); with $f(n)=1$ Permalink: http://dlmf.nist.gov/25.2.E7 Encodings: TeX, pMML, png See also: Annotations for §25.2(ii), §25.2 and Ch.25

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

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

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Keywords:: Riemann zeta function, representations by Euler–Maclaurin formula
Permalink:: http://dlmf.nist.gov/25.2.iii
See also:: Annotations for §25.2 and Ch.25

25.2.8		$\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N% }^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\,\mathrm{d}x,$
$\Re s>0$ , $N=1,2,3,\dots$ .
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\int$ : integral, $\Re$ : real part, $k$ : nonnegative integer, $x$ : real variable, $a$ : real or complex parameter and $s$ : complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Apostol (1976, (25), p. 269) Proof sketch: Derivable from Apostol (1976, Theorem 12.21, p. 269) by setting $a=1$ and $N\mapsto N-1$ . A&S Ref: 23.2.9 Referenced by: (25.2.9) Permalink: http://dlmf.nist.gov/25.2.E8 Encodings: TeX, pMML, png See also: Annotations for §25.2(iii), §25.2 and Ch.25

25.2.9		$\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$
$\Re s>-2n$ ; $n,N=1,2,3,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable Keywords: Euler–Maclaurin formula, improper integral Proof sketch: Derivable from (25.2.8) by repeated integration by parts. Referenced by: §25.18(i), (25.2.10) Permalink: http://dlmf.nist.gov/25.2.E9 Encodings: TeX, pMML, png See also: Annotations for §25.2(iii), §25.2 and Ch.25

25.2.10		$\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,$
$\Re s>-2n$ , $n=1,2,3,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable Keywords: Euler–Maclaurin formula Proof sketch: Derivable from (25.2.9) with $N=1$ . A&S Ref: 23.2.3 Permalink: http://dlmf.nist.gov/25.2.E10 Encodings: TeX, pMML, png See also: Annotations for §25.2(iii), §25.2 and Ch.25

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

§25.2(iv) Infinite Products

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Keywords:: Riemann zeta function, infinite products
Notes:: See Apostol (1976, Section 11.5) and Titchmarsh (1986b, Section 2.12).
Permalink:: http://dlmf.nist.gov/25.2.iv
See also:: Annotations for §25.2 and Ch.25

25.2.11		$\zeta\left(s\right)=\prod_{p}(1-p^{-s})^{-1},$
$\Re s>1$ ,
ⓘ Symbols: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\Re$ : real part, $p$ : prime number and $s$ : complex variable Keywords: infinite product, primes Source: Apostol (1976, p. 231) A&S Ref: 23.2.2 Referenced by: §25.10(i) Permalink: http://dlmf.nist.gov/25.2.E11 Encodings: TeX, pMML, png See also: Annotations for §25.2(iv), §25.2 and Ch.25

product over all primes $p$ .

25.2.12		$\zeta\left(s\right)=\frac{(2\pi)^{s}e^{-s-(\gamma s/2)}}{2(s-1)\Gamma\left(% \tfrac{1}{2}s+1\right)}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $s$ : complex variable and $\rho$ : zeros Keywords: infinite product, primes Source: Titchmarsh (1986b, (2.12.6), (2.12.8), p. 30–31) A&S Ref: 23.2.10 (in slightly different form) Permalink: http://dlmf.nist.gov/25.2.E12 Encodings: TeX, pMML, png See also: Annotations for §25.2(iv), §25.2 and Ch.25

product over zeros $\rho$ of $\zeta$ with $\Re\rho>0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).

25.1 Special Notation 25.3 Graphics

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§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