§25.5 Integral Representations

Keywords:: Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.5

§25.5(i) In Terms of Elementary Functions
§25.5(ii) In Terms of Other Functions
§25.5(iii) Contour Integrals

§25.5(i) In Terms of Elementary Functions

Notes:: See Apostol (1976, p. 251) and Erdélyi et al. (1953a, p. 32). For (25.5.2) and (25.5.4) integrate (25.5.1) and (25.5.3) by parts. For (25.5.5) see Titchmarsh (1986b, p. 15). (25.5.6) comes from (25.5.1) by using the identity $e^{{-x}}=(1-e^{{-x}})/(e^{x}-1)$ in the integral $\mathop{\Gamma\/}\nolimits\!\left(s\right)=\int _{0}^{{\infty}}e^{{-x}}x^{{s-1}}dx$ together with (5.5.1). (25.5.7) follows from (25.5.6) because $\mathop{\Gamma\/}\nolimits\!\left(s+2m-1\right)=\int _{0}^{{\infty}}e^{{-x}}x^{{s+2m-2}}dx$ . For (25.5.10) and (25.5.11) see Lindelöf (1905, p. 103). For (25.5.12) see Srivastava and Choi (2001, p. 12).
Permalink:: http://dlmf.nist.gov/25.5.i

Throughout this subsection $s\neq 1$ .

25.5.1 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\frac{x^{{s-1}}}{e^{x}-1}dx,$ $\realpart{s}>1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
A&S Ref:: 23.2.7
Referenced by:: ¶ ‣ §25.12(ii), §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E1
Encodings:: TeX, pMML, png

25.5.2 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s+1\right)}\int _{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}dx,$ $\realpart{s}>1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E2
Encodings:: TeX, pMML, png

25.5.3 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{(1-2^{{1-s}})\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\frac{x^{{s-1}}}{e^{x}+1}dx,$ $\realpart{s}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
A&S Ref:: 23.2.8
Referenced by:: §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E3
Encodings:: TeX, pMML, png

25.5.4 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{(1-2^{{1-s}})\mathop{\Gamma\/}\nolimits\!\left(s+1\right)}\int _{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}dx,$ $\realpart{s}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E4
Encodings:: TeX, pMML, png

25.5.5 $\mathop{\zeta\/}\nolimits\!\left(s\right)=-s\int _{0}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{x^{{s+1}}}dx,$ $-1<\realpart{s}<0$ .

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, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E5
Encodings:: TeX, pMML, png

25.5.6 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{{s-1}}}{e^{x}}dx,$ $\realpart{s}>-1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.11(vii), §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E6
Encodings:: TeX, pMML, png

25.5.7 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum _{{m=1}}^{n}\frac{\mathop{B_{{2m}}\/}\nolimits}{(2m)!}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+2m-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}+\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum _{{m=1}}^{n}\frac{\mathop{B_{{2m}}\/}\nolimits}{(2m)!}x^{{2m-1}}\right)\frac{x^{{s-1}}}{e^{x}}dx,$ $\realpart{s}>-(2n+1)$ , $n=1,2,3,\dots$ .

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral, $\realpart{}$ : real part, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.11(vii), §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E7
Encodings:: TeX, pMML, png

25.5.8 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{2(1-2^{{-s}})\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int _{0}^{\infty}\frac{x^{{s-1}}}{\mathop{\sinh\/}\nolimits x}dx,$ $\realpart{s}>1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.5.E8
Encodings:: TeX, pMML, png

25.5.9 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{2^{{s-1}}}{\mathop{\Gamma\/}\nolimits\!\left(s+1\right)}\int _{0}^{\infty}\frac{x^{s}}{(\mathop{\sinh\/}\nolimits x)^{2}}dx,$ $\realpart{s}>1$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.5.E9
Encodings:: TeX, pMML, png

25.5.10 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{2^{{s-1}}}{1-2^{{1-s}}}\int _{0}^{\infty}\frac{\mathop{\cos\/}\nolimits\!\left(s\mathop{\mathrm{arctan}\/}\nolimits x\right)}{(1+x^{2})^{{s/2}}\mathop{\cosh\/}\nolimits\!\left(\frac{1}{2}\pi x\right)}dx.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E10
Encodings:: TeX, pMML, png

25.5.11 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2\int _{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(s\mathop{\mathrm{arctan}\/}\nolimits x\right)}{(1+x^{2})^{{s/2}}(e^{{2\pi x}}-1)}dx.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E11
Encodings:: TeX, pMML, png

25.5.12 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{2^{{s-1}}}{s-1}-2^{s}\int _{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(s\mathop{\mathrm{arctan}\/}\nolimits x\right)}{(1+x^{2})^{{s/2}}(e^{{\pi x}}+1)}dx.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(i)
Permalink:: http://dlmf.nist.gov/25.5.E12
Encodings:: TeX, pMML, png

§25.5(ii) In Terms of Other Functions

Notes:: See de Bruijn (1937). For (25.5.13) see Titchmarsh (1986b, p. 22).
Keywords:: Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.5.ii

25.5.13 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{\pi^{{s/2}}}{s(s-1)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}s\right)}+\frac{\pi^{{s/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}s\right)}\*\int _{1}^{\infty}\left(x^{{s/2}}+x^{{(1-s)/2}}\right)\frac{\omega(x)}{x}dx,$ $s\neq 1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\int$ : integral, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(ii)
Permalink:: http://dlmf.nist.gov/25.5.E13
Encodings:: TeX, pMML, png

where

25.5.14 $\omega(x)=\sum _{{n=1}}^{\infty}e^{{-n^{2}\pi x}}=\frac{1}{2}\left(\mathop{\theta _{{3}}\/}\nolimits\!\left(0\middle|ix\right)-1\right).$

Symbols:: $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $e$ : base of exponential function, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/25.5.E14
Encodings:: TeX, pMML, png

For $\mathop{\theta _{{3}}\/}\nolimits$ see §20.2(i). For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339).

In (25.5.15)–(25.5.19), $0<\realpart{s}<1$ , $\mathop{\psi\/}\nolimits\!\left(x\right)$ is the digamma function, and $\EulerConstant$ is Euler’s constant (§5.2). (25.5.16) is also valid for $0<\realpart{s}<2$ , $s\neq 1$ .

25.5.15 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{s-1}+\frac{\mathop{\sin\/}\nolimits\!\left(\pi s\right)}{\pi}\*\int _{0}^{\infty}(\mathop{\ln\/}\nolimits\!\left(1+x\right)-\mathop{\psi\/}\nolimits\!\left(1+x\right))x^{{-s}}dx,$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(ii)
Permalink:: http://dlmf.nist.gov/25.5.E15
Encodings:: TeX, pMML, png

25.5.16 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{1}{s-1}+\frac{\mathop{\sin\/}\nolimits\!\left(\pi s\right)}{\pi(s-1)}\*\int _{0}^{\infty}\left(\frac{1}{1+x}-{\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(1+x\right)\right)x^{{1-s}}dx,$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(ii)
Permalink:: http://dlmf.nist.gov/25.5.E16
Encodings:: TeX, pMML, png

25.5.17 $\mathop{\zeta\/}\nolimits\!\left(1+s\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\pi s\right)}{\pi}\int _{0}^{\infty}\left(\EulerConstant+\mathop{\psi\/}\nolimits\!\left(1+x\right)\right)x^{{-s-1}}dx,$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.5.E17
Encodings:: TeX, pMML, png

25.5.18 $\mathop{\zeta\/}\nolimits\!\left(1+s\right)=\frac{\mathop{\sin\/}\nolimits\!\left(\pi s\right)}{\pi s}\int _{0}^{\infty}{\mathop{\psi\/}\nolimits^{{\prime}}}\!\left(1+x\right)x^{{-s}}dx,$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $s$ : complex variable
Permalink:: http://dlmf.nist.gov/25.5.E18
Encodings:: TeX, pMML, png

25.5.19 $\mathop{\zeta\/}\nolimits\!\left(m+s\right)=(-1)^{{m-1}}\frac{\mathop{\Gamma\/}\nolimits\!\left(s\right)\mathop{\sin\/}\nolimits\!\left(\pi s\right)}{\pi\mathop{\Gamma\/}\nolimits\!\left(m+s\right)}\*\int _{0}^{\infty}{\mathop{\psi\/}\nolimits^{{(m)}}}\!\left(1+x\right)x^{{-s}}dx,$ $m=1,2,3,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $m$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Referenced by:: §25.5(ii)
Permalink:: http://dlmf.nist.gov/25.5.E19
Encodings:: TeX, pMML, png

§25.5(iii) Contour Integrals

Notes:: See Apostol (1976, p. 253). For (25.5.21) see Erdélyi et al. (1953a, p. 32).
Keywords:: Riemann zeta function
Permalink:: http://dlmf.nist.gov/25.5.iii

25.5.20 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-s\right)}{2\pi i}\int _{{-\infty}}^{{(0+)}}\frac{z^{{s-1}}}{e^{{-z}}-1}dz,$ $s\neq 1,2,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $s$ : complex variable and $z$ : complex variable
A&S Ref:: 23.2.4 (in slightly different form)
Referenced by:: §25.11(vii)
Permalink:: http://dlmf.nist.gov/25.5.E20
Encodings:: TeX, pMML, png

where the integration contour is a loop around the negative real axis; it starts at $-\infty$ , encircles the origin once in the positive direction without enclosing any of the points $z=\pm 2\pi i$ , $\pm 4\pi i$ , …, and returns to $-\infty$ . Equivalently,

25.5.21 $\mathop{\zeta\/}\nolimits\!\left(s\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-s\right)}{2\pi i(1-2^{{1-s}})}\*\int _{{-\infty}}^{{(0+)}}\frac{z^{{s-1}}}{e^{{-z}}+1}dz,$ $s\neq 1,2,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $s$ : complex variable and $z$ : complex variable
Referenced by:: §25.5(iii)
Permalink:: http://dlmf.nist.gov/25.5.E21
Encodings:: TeX, pMML, png

The contour here is any loop that encircles the origin in the positive direction not enclosing any of the points $\pm\pi i$ , $\pm 3\pi i$ , ….

§25.5 Integral Representations

Contents

§25.5(i) In Terms of Elementary Functions

§25.5(ii) In Terms of Other Functions

§25.5(iii) Contour Integrals