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25 Zeta and Related FunctionsRelated Functions

§25.11 Hurwitz Zeta Function


§25.11(i) Definition

The function ζ(s,a) was introduced in Hurwitz (1882) and defined by the series expansion

25.11.1 ζ(s,a)=n=01(n+a)s,
s>1, a0,-1,-2,.

ζ(s,a) has a meromorphic continuation in the s-plane, its only singularity in being a simple pole at s=1 with residue 1. As a function of a, with s (1) fixed, ζ(s,a) is analytic in the half-plane a>0. The Riemann zeta function is a special case:

25.11.2 ζ(s,1)=ζ(s).

For most purposes it suffices to restrict 0<a1 because of the following straightforward consequences of (25.11.1):

25.11.3 ζ(s,a)=ζ(s,a+1)+a-s,
25.11.4 ζ(s,a)=ζ(s,a+m)+n=0m-11(n+a)s,

Most references treat real a with 0<a1.

§25.11(ii) Graphics

See accompanying text
Figure 25.11.1: Hurwitz zeta function ζ(x,a), a = 0.3, 0.5, 0.8, 1, -20x10. The curves are almost indistinguishable for -14<x<-1, approximately. Magnify 3D Help
See accompanying text
Figure 25.11.2: Hurwitz zeta function ζ(x,a), -19.5x10, 0.02a1. Magnify 3D Help

§25.11(iii) Representations by the Euler–Maclaurin Formula

25.11.5 ζ(s,a)=n=0N1(n+a)s+(N+a)1-ss-1-sNx-x(x+a)s+1dx,
s1, s>0, a>0, N=0,1,2,3,.
25.11.6 ζ(s,a)=1as(12+as-1)-s(s+1)20B~2(x)-B2(x+a)s+2dx,
s1, s>-1, a>0.
25.11.7 ζ(s,a)=1as+1(1+a)s(12+1+as-1)+k=1n(s+2k-22k-1)B2k2k1(1+a)s+2k-1-(s+2n2n+1)1B~2n+1(x)(x+a)s+2n+1dx,
s1, a>0, n=1,2,3,, s>-2n.

For B~n(x) see §24.2(iii).

§25.11(iv) Series Representations

25.11.8 ζ(s,12a)=ζ(s,12a+12)+2sn=0(-1)n(n+a)s,
s>0, s1, 0<a1.
25.11.9 ζ(1-s,a)=2Γ(s)(2π)sn=11nscos(12πs-2nπa),
s>0 if 0<a<1; s>1 if a=1.
25.11.10 ζ(s,a)=n=0(s)nn!ζ(n+s)(1-a)n,
s1, |a-1|<1.

When a=12, (25.11.10) reduces to (25.8.3); compare (25.11.11).

For other series expansions similar to (25.11.10) see Coffey (2008).

§25.11(v) Special Values

Throughout this subsection a>0.

25.11.11 ζ(s,12)=(2s-1)ζ(s),
25.11.12 ζ(n+1,a)=(-1)n+1ψ(n)(a)n!,
25.11.13 ζ(0,a)=12-a.
25.11.14 ζ(-n,a)=-Bn+1(a)n+1,
25.11.15 ζ(s,ka)=k-sn=0k-1ζ(s,a+nk),
s1, k=1,2,3,.
25.11.16 ζ(1-s,hk)=2Γ(s)(2πk)sr=1kcos(πs2-2πrhk)ζ(s,rk),
s0,1; h,k integers, 1hk.

§25.11(vi) Derivatives


25.11.17 aζ(s,a)=-sζ(s+1,a),
s0,1; a>0.


In (25.11.18)–(25.11.24) primes on ζ denote derivatives with respect to s. Similarly in §§25.11(viii) and 25.11(xii).

25.11.18 ζ(0,a)=lnΓ(a)-12ln(2π),
25.11.19 ζ(s,a)=-lnaas(12+as-1)-a1-s(s-1)2+s(s+1)20(B~2(x)-B2)ln(x+a)(x+a)s+2dx-(2s+1)20B~2(x)-B2(x+a)s+2dx,
s>-1, s1, a>0.
25.11.20 (-1)kζ(k)(s,a)=(lna)kas(12+as-1)+k!a1-sr=0k-1(lna)rr!(s-1)k-r+1-s(s+1)20(B~2(x)-B2)(ln(x+a))k(x+a)s+2dx+k(2s+1)20(B~2(x)-B2)(ln(x+a))k-1(x+a)s+2dx-k(k-1)20(B~2(x)-B2)(ln(x+a))k-2(x+a)s+2dx,
s>-1, s1, a>0.
25.11.21 ζ(1-2n,hk)=(ψ(2n)-ln(2πk))B2n(h/k)2n-(ψ(2n)-ln(2π))B2n2nk2n+(-1)n+1π(2πk)2nr=1k-1sin(2πrhk)ψ(2n-1)(rk)+(-1)n+12(2n-1)!(2πk)2nr=1k-1cos(2πrhk)ζ(2n,rk)+ζ(1-2n)k2n,

where h,k are integers with 1hk and n=1,2,3,.

25.11.22 ζ(1-2n,12)=-B2nln2n4n-(22n-1-1)ζ(1-2n)22n-1,
25.11.23 ζ(1-2n,13)=-π(9n-1)B2n8n3(32n-1-1)-B2nln34n32n-1-(-1)nψ(2n-1)(13)23(6π)2n-1-(32n-1-1)ζ(1-2n)232n-1,
25.11.24 r=1k-1ζ(s,rk)=(ks-1)ζ(s)+ksζ(s)lnk,
s1, k=1,2,3,.

§25.11(vii) Integral Representations

25.11.25 ζ(s,a) =1Γ(s)0xs-1e-ax1-e-xdx,
s>1, a>0.
25.11.26 ζ(s,a) =-s-ax-x-12(x+a)s+1dx,
-1<s<0, 0<a1.
25.11.27 ζ(s,a)=12a-s+a1-ss-1+1Γ(s)0(1ex-1-1x+12)xs-1eaxdx,
s>-1, s1, a>0.
25.11.28 ζ(s,a)=12a-s+a1-ss-1+k=1nB2k(2k)!(s)2k-1a1-s-2k+1Γ(s)0(1ex-1-1x+12-k=1nB2k(2k)!x2k-1)xs-1e-axdx,
s>-(2n+1), s1, a>0.
25.11.29 ζ(s,a)=12a-s+a1-ss-1+20sin(sarctan(x/a))(a2+x2)s/2(e2πx-1)dx,
s1, a>0.
25.11.30 ζ(s,a)=Γ(1-s)2πi-(0+)eazzs-11-ezdz,
s1, a>0,

where the integration contour is a loop around the negative real axis as described for (25.5.20).

§25.11(viii) Further Integral Representations

25.11.31 1Γ(s)0xs-1e-ax2coshxdx=4-s(ζ(s,14+14a)-ζ(s,34+14a)),
s>0, a>-1.
25.11.32 0axnψ(x)dx=(-1)n-1ζ(-n)+(-1)nHnBn+1n+1-k=0n(-1)k(nk)HkBk+1(a)k+1an-k+k=0n(-1)k(nk)ζ(-k,a)an-k,
n=1,2,, a>0,

where Hn are the harmonic numbers:

25.11.33 Hn=k=1nk-1.
25.11.34 n0aζ(1-n,x)dx=ζ(-n,a)-ζ(-n)+Bn+1-Bn+1(a)n(n+1),
n=1,2,, a>0.

§25.11(ix) Integrals

See Prudnikov et al. (1990, §2.3), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).

§25.11(x) Further Series Representations

25.11.35 n=0(-1)n(n+a)s=1Γ(s)0xs-1e-ax1+e-xdx=2-s(ζ(s,12a)-ζ(s,12(1+a))),
a>0, s>0; or a=0, a0, 0<s<1.

When a=1, (25.11.35) reduces to (25.2.3).

25.11.36 Removed because it is just (25.15.1) combined with (25.15.3).

See also (8.15.2) and Srivastava and Choi (2001).

§25.11(xi) Sums

25.11.37 k=1(-1)kkζ(nk,a)=-nlnΓ(a)+ln(j=0n-1Γ(a-e(2j+1)πi/n)),
n=2,3,4,, a1.
25.11.38 k=1(n+kk)ζ(n+k+1,a)zk=(-1)nn!(ψ(n)(a)-ψ(n)(a-z)),
n=1,2,3,, a>0, |z|<|a|.
25.11.39 k=2k2kζ(k+1,34)=8G,

where G is Catalan’s constant:

25.11.40 Gn=0(-1)n(2n+1)2=0.91596 55941 772.

For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360).

§25.11(xii) a-Asymptotic Behavior

As a0 with s (1) fixed,

25.11.41 ζ(s,a+1)=ζ(s)-sζ(s+1)a+O(a2).

As β± with s fixed, s>1,

25.11.42 ζ(s,α+iβ)0,

uniformly with respect to bounded nonnegative values of α.

As a in the sector |pha|π-δ(<π), with s(1) and δ fixed, we have the asymptotic expansion

25.11.43 ζ(s,a)-a1-ss-1-12a-sk=1B2k(2k)!(s)2k-1a1-s-2k.

Similarly, as a in the sector |pha|12π-δ(<12π),

25.11.44 ζ(-1,a)-112+14a2-(112-12a+12a2)lna-k=1B2k+2(2k+2)(2k+1)2ka-2k,


25.11.45 ζ(-2,a)-112a+19a3-(16a-12a2+13a3)lnak=12B2k+2(2k+2)(2k+1)2k(2k-1)a-(2k-1).

For the more general case ζ(-m,a), m=1,2,, see Elizalde (1986).

For an exponentially-improved form of (25.11.43) see Paris (2005b).