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25 Zeta and Related FunctionsRiemann Zeta Function

§25.2 Definition and Expansions

  1. §25.2(i) Definition
  2. §25.2(ii) Other Infinite Series
  3. §25.2(iii) Representations by the Euler–Maclaurin Formula
  4. §25.2(iv) Infinite Products

§25.2(i) Definition

When s>1,

25.2.1 ζ(s)=n=11ns.

Elsewhere ζ(s) is defined by analytic continuation. It is a meromorphic function whose only singularity in is a simple pole at s=1, with residue 1.

§25.2(ii) Other Infinite Series

25.2.2 ζ(s)=112sn=01(2n+1)s,
25.2.3 ζ(s)=1121sn=1(1)n1ns,
25.2.4 ζ(s)=1s1+n=0(1)nn!γn(s1)n,

where the Stieltjes constants γn are defined via

25.2.5 γn=limm(k=1m(lnk)nk(lnm)n+1n+1).
25.2.6 ζ(s)=n=2(lnn)ns,
25.2.7 ζ(k)(s)=(1)kn=2(lnn)kns,
s>1, k=1,2,3,.

For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. This includes, for example, 1/ζ(s).

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

25.2.8 ζ(s)=k=1N1ks+N1ss1sNxxxs+1dx,
s>0, N=1,2,3,.
25.2.9 ζ(s)=k=1N1ks+N1ss112Ns+k=1n(s+2k22k1)B2k2kN1s2k(s+2n2n+1)NB~2n+1(x)xs+2n+1dx,
s>2n; n,N=1,2,3,.
25.2.10 ζ(s)=1s1+12+k=1n(s+2k22k1)B2k2k(s+2n2n+1)1B~2n+1(x)xs+2n+1dx,
s>2n, n=1,2,3,.

For B2k see §24.2(i), and for B~n(x) see §24.2(iii).

§25.2(iv) Infinite Products

25.2.11 ζ(s)=p(1ps)1,

product over all primes p.

25.2.12 ζ(s)=(2π)ses(γs/2)2(s1)Γ(12s+1)ρ(1sρ)es/ρ,

product over zeros ρ of ζ with ρ>0 (see §25.10(i)); γ is Euler’s constant (§5.2(ii)).