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25 Zeta and Related FunctionsRiemann Zeta Function

§25.2 Definition and Expansions

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§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)=11-2-sn=01(2n+1)s,
s>1.
25.2.3 ζ(s)=11-21-sn=1(-1)n-1ns,
s>0.
25.2.4 ζ(s)=1s-1+n=0(-1)nn!γn(s-1)n,
s>0,

where

25.2.5 γn=limm(k=1m(lnk)nk-(lnm)n+1n+1).
25.2.6 ζ(s)=-n=2(lnn)n-s,
s>1.
25.2.7 ζ(k)(s)=(-1)kn=2(lnn)kn-s,
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+N1-ss-1-sNx-xxs+1dx,
s>0, N=1,2,3,.
25.2.9 ζ(s)=k=1N1ks+N1-ss-1-12N-s+k=1n(s+2k-22k-1)B2k2kN1-s-2k-(s+2n2n+1)NB~2n+1(x)xs+2n+1dx,
s>-2n; n,N=1,2,3,.
25.2.10 ζ(s)=1s-1+12+k=1n(s+2k-22k-1)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(1-p-s)-1,
s>1,

product over all primes p.

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

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