Digital Library of Mathematical Functions
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-1x-1x,
s>1.
25.5.2 ζ(s) =1Γ(s+1)0xxs(x-1)2x,
s>1.
25.5.3 ζ(s) =1(1-21-s)Γ(s)0xs-1x+1x,
s>0.
25.5.4 ζ(s) =1(1-21-s)Γ(s+1)0xxs(x+1)2x,
s>0.
25.5.5 ζ(s)=-s0x-x-12xs+1x,
-1<s<0.
25.5.6 ζ(s)=12+1s-1+1Γ(s)0(1x-1-1x+12)xs-1xx,
s>-1.
25.5.7 ζ(s)=12+1s-1+m=1nB2m(2m)!Γ(s+2m-1)Γ(s)+1Γ(s)0(1x-1-1x+12-m=1nB2m(2m)!x2m-1)xs-1xx,
s>-(2n+1), n=1,2,3,.
25.5.8 ζ(s) =12(1-2-s)Γ(s)0xs-1sinhxx,
s>1.
25.5.9 ζ(s) =2s-1Γ(s+1)0xs(sinhx)2x,
s>1.
25.5.10 ζ(s)=2s-11-21-s0cos(sarctanx)(1+x2)s/2cosh(12πx)x.
25.5.11 ζ(s)=12+1s-1+20sin(sarctanx)(1+x2)s/2(2πx-1)x.
25.5.12 ζ(s)=2s-1s-1-2s0sin(sarctanx)(1+x2)s/2(πx+1)x.

§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)xx,
s1,

where

25.5.14 ω(x)=n=1-n2πx=12(θ3(0|x)-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-sx,
25.5.16 ζ(s) =1s-1+sin(πs)π(s-1)0(11+x-ψ(1+x))x1-sx,
25.5.17 ζ(1+s) =sin(πs)π0(γ+ψ(1+x))x-s-1x,
25.5.18 ζ(1+s) =sin(πs)πs0ψ(1+x)x-sx,
25.5.19 ζ(m+s) =(-1)m-1Γ(s)sin(πs)πΓ(m+s)0ψ(m)(1+x)x-sx,
m=1,2,3,.

§25.5(iii) Contour Integrals

25.5.20 ζ(s)=Γ(1-s)2π-(0+)zs-1-z-1z,
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=±2π, ±4π, …, and returns to -. Equivalently,

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

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