§25.2 Definition and Expansions

When $\mathrm{\Re }s>1$,

25.2.1		$$\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}.$$
ⓘ Defines: $\zeta \left(s\right)$: 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: (25.2.2), (25.2.4), §25.2(ii), §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 $ℂ$ is a simple pole at $s=1$, with residue 1.

25.2.2		$$\zeta \left(s\right)=\frac{1}{1-2^{-s}}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(2n+1)}^s},$$
$\mathrm{\Re }s>1$.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable Keywords: infinite series Source: 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), §25.11(v), §25.2(ii) 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		$$\zeta \left(s\right)=\frac{1}{1-2^{1-s}}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^{n-1}}{n^s},$$
$\mathrm{\Re }s>0$.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: 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), §25.2(ii) 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		$$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: 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: §25.2(ii), 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 $\mathrm{\Re }s>0$. 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		$${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$
ⓘ Defines: ${\gamma }_n$: Stieltjes constants Symbols: $\ln z$: 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		$${\zeta }^{\prime }\left(s\right)=-\sum _{n=2}^{\mathrm{\infty }}(\ln n)n^{-s},$$
$\mathrm{\Re }s>1$.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable Keywords: infinite series Source: Apostol (1976, (12), p. 236) Referenced by: §25.2(ii) 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		$${\zeta }^{(k)}\left(s\right)={(-1)}^k\sum _{n=2}^{\mathrm{\infty }}{(\ln n)}^k{n}^{-s},$$
$\mathrm{\Re }s>1$, $k=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $s$: complex variable Keywords: infinite series Source: Apostol (1976, p. 236); with $f(n)=1$ Referenced by: §25.2(ii) 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, $1/\zeta \left(s\right)$.

25.2.8		$$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{x^{s+1}}dx,$$
$\mathrm{\Re }s>0$, $N=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$, $\lfloor x\rfloor$: floor of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Apostol (1976, p. 269) A&S Ref: 23.2.9 Referenced by: (25.2.9), §25.2(iii) 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		$$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$
$\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$, $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Derivable from (25.2.8) by repeated integration by parts. Referenced by: §25.18(i), (25.2.10), §25.2(iii) 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		$$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$
$\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$, $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable Keywords: Euler–Maclaurin formula Source: 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 ${\tilde{B}}_n\left(x\right)$ see §24.2(iii).

25.2.11		$$\zeta \left(s\right)=\prod _p{(1-p^{-s})}^{-1},$$
$\mathrm{\Re }s>1$,
ⓘ Symbols: $\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: 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		$$\zeta \left(s\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{-s-(\gamma s/2)}}{2(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s+1\right)}\prod _{\rho }\left(1-\frac{s}{\rho }\right){\mathrm{e}}^{s/\rho },$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: 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, 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 $\mathrm{\Re }\rho >0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).