# §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 A&S Ref: 23.2.1 Referenced by: §25.2(ii), §25.2(ii) Permalink: http://dlmf.nist.gov/25.2.E1 Encodings: TeX, pMML, png See also: Annotations for 25.2(i), 25.2 and 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 A&S Ref: 23.2.20 (is the special case with integer values of $s$) Referenced by: §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 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 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 25
 25.2.4 ${\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}% \gamma_{n}(s-1)^{n},}$ $\Re s>0$, ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $\Re$: real part, $n$: nonnegative integer, $s$: complex variable and $\gamma_{n}$: expansion coefficient A&S Ref: 23.2.5 Referenced by: §25.2(ii) Permalink: http://dlmf.nist.gov/25.2.E4 Encodings: TeX, pMML, png See also: Annotations for 25.2(ii), 25.2 and 25

where

 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: $\gamma_{n}$: expansion coefficient (locally) Symbols: $\ln\NVar{z}$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §25.2(ii), §25.6(ii) Permalink: http://dlmf.nist.gov/25.2.E5 Encodings: TeX, pMML, png See also: Annotations for 25.2(ii), 25.2 and 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 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 25
 25.2.7 ${\zeta^{(k)}}\left(s\right)=(-1)^{k}\sum_{n=2}^{\infty}(\ln n)^{k}n^{-s},$ $\Re s>1$, $k=1,2,3,\dots$.

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 $\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N% }^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\mathrm{d}x,$ $\Re s>0$, $N=1,2,3,\dots$.
 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 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 25

product over all primes $p$.

 25.2.12 $\zeta\left(s\right)=\frac{(2\pi)^{s}e^{-s-(\gamma s/2)}}{2(s-1)\Gamma\left(% \tfrac{1}{2}s+1\right)}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho},$

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