What's New
About the Project
NIST
25 Zeta and Related FunctionsRiemann Zeta Function

§25.5 Integral Representations

Contents

§25.5(i) In Terms of Elementary Functions

Throughout this subsection s1.

25.5.1 ζ(s) =1Γ(s)0xs-1ex-1dx,
s>1.
25.5.2 ζ(s) =1Γ(s+1)0exxs(ex-1)2dx,
s>1.
25.5.3 ζ(s) =1(1-21-s)Γ(s)0xs-1ex+1dx,
s>0.
25.5.4 ζ(s) =1(1-21-s)Γ(s+1)0exxs(ex+1)2dx,
s>0.
25.5.5 ζ(s)=-s0x-x-12xs+1dx,
-1<s<0.
25.5.6 ζ(s)=12+1s-1+1Γ(s)0(1ex-1-1x+12)xs-1exdx,
s>-1.
25.5.7 ζ(s)=12+1s-1+m=1nB2m(2m)!Γ(s+2m-1)Γ(s)+1Γ(s)0(1ex-1-1x+12-m=1nB2m(2m)!x2m-1)xs-1exdx,
s>-(2n+1), n=1,2,3,.
25.5.8 ζ(s) =12(1-2-s)Γ(s)0xs-1sinhxdx,
s>1.
25.5.9 ζ(s) =2s-1Γ(s+1)0xs(sinhx)2dx,
s>1.
25.5.10 ζ(s)=2s-11-21-s0cos(sarctanx)(1+x2)s/2cosh(12πx)dx.
25.5.11 ζ(s)=12+1s-1+20sin(sarctanx)(1+x2)s/2(e2πx-1)dx.
25.5.12 ζ(s)=2s-1s-1-2s0sin(sarctanx)(1+x2)s/2(eπx+1)dx.

§25.5(ii) In Terms of Other Functions

25.5.13 ζ(s)=πs/2s(s-1)Γ(12s)+πs/2Γ(12s)1(xs/2+x(1-s)/2)ω(x)xdx,
s1,

where

25.5.14 ω(x)=n=1e-n2πx=12(θ3(0|ix)-1).

For θ3 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<s<1, ψ(x) is the digamma function, and γ is Euler’s constant (§5.2). (25.5.16) is also valid for 0<s<2, s1.

25.5.15 ζ(s)=1s-1+sin(πs)π0(ln(1+x)-ψ(1+x))x-sdx,
25.5.16 ζ(s) =1s-1+sin(πs)π(s-1)0(11+x-ψ(1+x))x1-sdx,
25.5.17 ζ(1+s) =sin(πs)π0(γ+ψ(1+x))x-s-1dx,
25.5.18 ζ(1+s) =sin(πs)πs0ψ(1+x)x-sdx,
25.5.19 ζ(m+s) =(-1)m-1Γ(s)sin(πs)πΓ(m+s)0ψ(m)(1+x)x-sdx,
m=1,2,3,.

§25.5(iii) Contour Integrals

25.5.20 ζ(s)=Γ(1-s)2πi-(0+)zs-1e-z-1dz,
s1,2,,

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

25.5.21 ζ(s)=Γ(1-s)2πi(1-21-s)-(0+)zs-1e-z+1dz,
s1,2,.

The contour here is any loop that encircles the origin in the positive direction not enclosing any of the points ±πi, ±3πi, ….