Contents
- §25.2(i) Definition
- §25.2(ii) Other Infinite Series
- §25.2(iii) Representations by the Euler–Maclaurin Formula
- §25.2(iv) Infinite Products
§25.2(i) Definition
When ℜs>1,
Elsewhere ζ(s) is defined by analytic continuation. It is a meromorphic function whose only singularity in ℂ is a simple pole ats=1, with residue 1.
§25.2(ii) Other Infinite Series
| 25.2.4 |
ζ(s)=1s−1+∑n=0∞(−1)nn!γn(s−1)n, |
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where the Stieltjes constants γn are defined via
| 25.2.5 |
γn=limm→∞(∑k=1m(lnk)nk−(lnm)n+1n+1). |
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| 25.2.7 |
ζ(k)(s)=(−1)k∑n=2∞(lnn)kn−s, |
| ℜs>1, k=1,2,3,…. |
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| ⓘ Symbols: ζ(s): Riemann zeta function,lnz: principal branch of logarithm function,ℜ: real part,k: nonnegative integer,n: nonnegative integer ands: complex variable Keywords: infinite series Source: Apostol (1976, p. 236); with f(n)=1 Permalink: http://dlmf.nist.gov/25.2.E7 Encodings: TeX, pMML, png See also: Annotations for §25.2(ii),§25.2 andCh.25 |
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For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. This includes, for example, 1/ζ(s).
| 25.2.8 |
ζ(s)=∑k=1N1ks+N1−ss−1−s∫N∞x−⌊x⌋xs+1dx, |
| ℜs>0, N=1,2,3,…. |
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| ⓘ Symbols: ζ(s): Riemann zeta function,dx: differential of x,⌊x⌋: floor of x,∫: integral,ℜ: real part,k: nonnegative integer,x: real variable,a: real or complex parameter ands: complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Apostol (1976, (25), p. 269) Proof sketch: Derivable from Apostol (1976, Theorem 12.21, p. 269) by settinga=1 and N↦N−1. A&S Ref: 23.2.9 Referenced by: (25.2.9) Permalink: http://dlmf.nist.gov/25.2.E8 Encodings: TeX, pMML, png See also: Annotations for §25.2(iii),§25.2 andCh.25 |
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| 25.2.9 |
ζ(s)=∑k=1N1ks+N1−ss−1−12N−s+∑k=1n(s+2k−22k−1)B2k2kN1−s−2k−(s+2n2n+1)∫N∞B~2n+1(x)xs+2n+1dx, |
| ℜs>−2n; n,N=1,2,3,…. |
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| ⓘ Symbols: Bn: Bernoulli numbers,ζ(s): Riemann zeta function,(mn): binomial coefficient,dx: differential of x,∫: integral,B~n(x): periodic Bernoulli functions,ℜ: real part,k: nonnegative integer,n: nonnegative integer,x: real variable ands: complex variable Keywords: Euler–Maclaurin formula, improper integral Proof sketch: Derivable from (25.2.8) by repeated integration by parts. Referenced by: §25.18(i),(25.2.10) Permalink: http://dlmf.nist.gov/25.2.E9 Encodings: TeX, pMML, png See also: Annotations for §25.2(iii),§25.2 andCh.25 |
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| 25.2.10 |
ζ(s)=1s−1+12+∑k=1n(s+2k−22k−1)B2k2k−(s+2n2n+1)∫1∞B~2n+1(x)xs+2n+1dx, |
| ℜs>−2n, n=1,2,3,…. |
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| ⓘ Symbols: Bn: Bernoulli numbers,ζ(s): Riemann zeta function,(mn): binomial coefficient,dx: differential of x,∫: integral,B~n(x): periodic Bernoulli functions,ℜ: real part,k: nonnegative integer,n: nonnegative integer,x: real variable ands: complex variable Keywords: Euler–Maclaurin formula Proof sketch: Derivable from (25.2.9) with N=1. A&S Ref: 23.2.3 Permalink: http://dlmf.nist.gov/25.2.E10 Encodings: TeX, pMML, png See also: Annotations for §25.2(iii),§25.2 andCh.25 |
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For B2k see §24.2(i), and for B~n(x)see §24.2(iii).
§25.2(iv) Infinite Products
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, |
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product over zeros ρ of ζ with ℜρ>0 (see §25.10(i)); γ is Euler’s constant (§5.2(ii)).
