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25 Zeta and Related FunctionsRelated Functions

§25.11 Hurwitz Zeta Function

  1. §25.11(i) Definition
  2. §25.11(ii) Graphics
  3. §25.11(iii) Representations by the Euler–Maclaurin Formula
  4. §25.11(iv) Series Representations
  5. §25.11(v) Special Values
  6. §25.11(vi) Derivatives
  7. §25.11(vii) Integral Representations
  8. §25.11(viii) Further Integral Representations
  9. §25.11(ix) Integrals
  10. §25.11(x) Further Series Representations
  11. §25.11(xi) Sums
  12. §25.11(xii) a-Asymptotic Behavior

§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)+as,
25.11.4 ζ(s,a)=ζ(s,a+m)+n=0m11(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)1ss1sNxx(x+a)s+1dx,
s1, s>0, a>0, N=0,1,2,3,.
25.11.6 ζ(s,a)=1as(12+as1)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+as1)+k=1n(s+2k22k1)B2k2k1(1+a)s+2k1(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(1s,a)=2Γ(s)(2π)sn=11nscos(12πs2nπa),
s>0 if 0<a<1; s>1 if a=1.
25.11.10 ζ(s,a)=n=0(s)nn!ζ(n+s)(1a)n,
s1, |a1|<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)=(2s1)ζ(s),
25.11.12 ζ(n+1,a)=(1)n+1ψ(n)(a)n!,
25.11.13 ζ(0,a)=12a.
25.11.14 ζ(n,a)=Bn+1(a)n+1,
25.11.15 ζ(s,ka)=ksn=0k1ζ(s,a+nk),
s1, k=1,2,3,.
25.11.16 ζ(1s,hk)=2Γ(s)(2πk)sr=1kcos(πs22πrhk)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+as1)a1s(s1)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+as1)+k!a1sr=0k1(lna)rr!(s1)kr+1s(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))k1(x+a)s+2dxk(k1)20(B~2(x)B2)(ln(x+a))k2(x+a)s+2dx,
s>1, s1, a>0.
25.11.21 ζ(12n,hk)=(ψ(2n)ln(2πk))B2n(h/k)2n(ψ(2n)ln(2π))B2n2nk2n+(1)n+1π(2πk)2nr=1k1sin(2πrhk)ψ(2n1)(rk)+(1)n+12(2n1)!(2πk)2nr=1k1cos(2πrhk)ζ(2n,rk)+ζ(12n)k2n,

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

25.11.22 ζ(12n,12)=B2nln2n4n(22n11)ζ(12n)22n1,
25.11.23 ζ(12n,13)=π(9n1)B2n8n3(32n11)B2nln34n32n1(1)nψ(2n1)(13)23(6π)2n1(32n11)ζ(12n)232n1,
25.11.24 r=1k1ζ(s,rk)=(ks1)ζ(s)+ksζ(s)lnk,
s1, k=1,2,3,.

§25.11(vii) Integral Representations

25.11.25 ζ(s,a) =1Γ(s)0xs1eax1exdx,
s>1, a>0.
25.11.26 ζ(s,a) =saxx12(x+a)s+1dx,
1<s<0, 0<a1.
25.11.27 ζ(s,a)=12as+a1ss1+1Γ(s)0(1ex11x+12)xs1eaxdx,
s>1, s1, a>0.
25.11.28 ζ(s,a)=12as+a1ss1+k=1nB2k(2k)!(s)2k1a1s2k+1Γ(s)0(1ex11x+12k=1nB2k(2k)!x2k1)xs1eaxdx,
s>(2n+1), s1, a>0.
25.11.29 ζ(s,a)=12as+a1ss1+20sin(sarctan(x/a))(a2+x2)s/2(e2πx1)dx,
s1, a>0.
25.11.30 ζ(s,a)=Γ(1s)2πi(0+)eazzs11ezdz,
s1, a>0,

where the integration contour (see Figure 5.9.1) 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)0xs1eax2coshxdx=4s(ζ(s,14+14a)ζ(s,34+14a)),
s>0, a>1.
25.11.32 0axnψ(x)dx=(1)n1ζ(n)+(1)nHnBn+1n+1k=0n(1)k(nk)HkBk+1(a)k+1ank+k=0n(1)k(nk)ζ(k,a)ank,
n=1,2,, a>0,

where Hn are the harmonic numbers:

25.11.33 Hn=k=1nk1.
25.11.34 n0aζ(1n,x)dx=ζ(n,a)ζ(n)+Bn+1Bn+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)0xs1eax1+exdx=2s(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=0n1Γ(ae(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)(az)),
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|πδ(<π), with s(1) and δ fixed, we have the asymptotic expansion

25.11.43 ζ(s,a)a1ss112ask=1B2k(2k)!(s)2k1a1s2k.

Similarly, as a in the sector |pha|πδ(<π).

25.11.44 ζ(1,a)112+14a2(11212a+12a2)lnak=1B2k+2(2k+2)(2k+1)2ka2k,


25.11.45 ζ(2,a)112a+19a3(16a12a2+13a3)lnak=12B2k+2(2k+2)(2k+1)2k(2k1)a(2k1).

For the more general case ζ(m,a), m=1,2,, see Elizalde (1986). For error bounds for (25.11.43), (25.11.44) and (25.11.45), see Nemes (2017a).

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