# §25.2 Definition and Expansions

## §25.2(i) Definition

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: (25.2.2), (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 $\mathbb{C}$ is a simple pole at $s=1$, with residue 1.

## §25.2(ii) Other Infinite Series

 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 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 $\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^% {s}},$ $\Re s>0$. ⓘ 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), §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 $\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$ ⓘ 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: §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 $\Re s>0$. 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) 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 ${\zeta}^{(k)}\left(s\right)=(-1)^{k}\sum_{n=2}^{\infty}(\ln n)^{k}n^{-s},$ $\Re s>1$, $k=1,2,3,\dots$. ⓘ 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 $f(n)=1$ 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(iii) Representations by the Euler–Maclaurin Formula

 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, $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$, $\int$: integral, $\Re$: 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 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\mathrm{d}x,$ $\Re s>-2n$; $n,N=1,2,3,\dots$.
 25.2.10 $\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\mathrm{d}x,$ $\Re s>-2n$, $n=1,2,3,\dots$.

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

## §25.2(iv) Infinite Products

 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, 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)).