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Riemann zeta function

Table of contents: Definitions - Illustrations - Dirichlet series - Euler product - Special values - Analytic properties - Zeros - Complex parts - Functional equation - Bounds and inequalities - Euler-Maclaurin formula - Approximations

Definitions

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Symbol: RiemannZeta ζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
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Symbol: RiemannZetaZero ρn\rho_{n} Nontrivial zero of the Riemann zeta function

Illustrations

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Image: X-ray of ζ ⁣(s)\zeta\!\left(s\right) on s[22,22]+[27,27]is \in \left[-22, 22\right] + \left[-27, 27\right] i with the critical strip highlighted

Dirichlet series

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ζ ⁣(s)=k=11ks\zeta\!\left(s\right) = \sum_{k=1}^{\infty} \frac{1}{{k}^{s}}
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1ζ ⁣(s)=k=1μ ⁣(k)ks\frac{1}{\zeta\!\left(s\right)} = \sum_{k=1}^{\infty} \frac{\mu\!\left(k\right)}{{k}^{s}}

Euler product

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ζ ⁣(s)=p111ps\zeta\!\left(s\right) = \prod_{p} \frac{1}{1 - \frac{1}{{p}^{s}}}

Special values

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ζ ⁣(2)=π26\zeta\!\left(2\right) = \frac{{\pi}^{2}}{6}
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ζ ⁣(3)Q\zeta\!\left(3\right) \notin \mathbb{Q}
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ζ ⁣(2n)=(1)n+1B2n(2π)2n2(2n)!\zeta\!\left(2 n\right) = \frac{{\left(-1\right)}^{n + 1} B_{2 n} {\left(2 \pi\right)}^{2 n}}{2 \left(2 n\right)!}
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ζ ⁣(n)=(1)nBn+1n+1\zeta\!\left(-n\right) = \frac{{\left(-1\right)}^{n} B_{n + 1}}{n + 1}
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Table of ζ ⁣(2n)\zeta\!\left(2 n\right) for 1n201 \le n \le 20
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Table of ζ ⁣(n)\zeta\!\left(-n\right) for 0n300 \le n \le 30
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Table of ζ ⁣(n)\zeta\!\left(n\right) to 50 digits for 2n502 \le n \le 50

Analytic properties

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HolomorphicDomain ⁣(ζ ⁣(s),s,C{~})=C{1}\operatorname{HolomorphicDomain}\!\left(\zeta\!\left(s\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \mathbb{C} \setminus \left\{1\right\}
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Poles ⁣(ζ ⁣(s),s,C{~})={1}\operatorname{Poles}\!\left(\zeta\!\left(s\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{1\right\}
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EssentialSingularities ⁣(ζ ⁣(s),s,C{~})={~}\operatorname{EssentialSingularities}\!\left(\zeta\!\left(s\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{{\tilde \infty}\right\}
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BranchPoints ⁣(ζ ⁣(s),s,C{~})={}\operatorname{BranchPoints}\!\left(\zeta\!\left(s\right), s, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchCuts ⁣(ζ ⁣(s),s,C)={}\operatorname{BranchCuts}\!\left(\zeta\!\left(s\right), s, \mathbb{C}\right) = \left\{\right\}

Zeros

Related topics: Zeros of the Riemann zeta function

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zerossRζ ⁣(s)={2n:nZ1}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{R}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\}
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zerossCζ ⁣(s)={2n:nZ1}{ρn:nZandn0}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C}} \zeta\!\left(s\right) = \left\{ -2 n : n \in \mathbb{Z}_{\ge 1} \right\} \cup \left\{ \rho_{n} : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \right\}
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0<Re ⁣(ρn)<10 \lt \operatorname{Re}\!\left(\rho_{n}\right) \lt 1
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Re ⁣(ρn)=12\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}
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ρn=ρn\rho_{-n} = \overline{\rho_{n}}
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Table of Im ⁣(ρn)\operatorname{Im}\!\left(\rho_{n}\right) to 50 digits for 1n501 \le n \le 50

Complex parts

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ζ ⁣(s)=ζ ⁣(s)\zeta\!\left(\overline{s}\right) = \overline{\zeta\!\left(s\right)}

Functional equation

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ζ ⁣(s)=2(2π)s1sin ⁣(πs2)Γ ⁣(1s)ζ ⁣(1s)\zeta\!\left(s\right) = 2 {\left(2 \pi\right)}^{s - 1} \sin\!\left(\frac{\pi s}{2}\right) \Gamma\!\left(1 - s\right) \zeta\!\left(1 - s\right)

Bounds and inequalities

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ζ ⁣(s)ζ ⁣(Re ⁣(s))\left|\zeta\!\left(s\right)\right| \le \zeta\!\left(\operatorname{Re}\!\left(s\right)\right)
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ζ ⁣(s)<31+s1s1+s2π(1+ηRe(s))/2ζ ⁣(1+η)\left|\zeta\!\left(s\right)\right| \lt 3 \left|\frac{1 + s}{1 - s}\right| {\left|\frac{1 + s}{2 \pi}\right|}^{\left( 1 + \eta - \operatorname{Re}\left(s\right) \right) / 2} \zeta\!\left(1 + \eta\right)

Euler-Maclaurin formula

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ζ ⁣(s)=k=1N11ks+N1ss1+1Ns(12+k=1MB2k(2k)!(s)2k1N2k1)NB2M ⁣(tt)(2M)!(s)2Mts+2Mdt\zeta\!\left(s\right) = \sum_{k=1}^{N - 1} \frac{1}{{k}^{s}} + \frac{{N}^{1 - s}}{s - 1} + \frac{1}{{N}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{N}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{t}^{s + 2 M}} \, dt

Approximations

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ζ ⁣(s)(k=1N11ks+N1ss1+1Ns(12+k=1MB2k(2k)!(s)2k1N2k1))4(s)2M(2π)2MN(Re ⁣(s)+2M1)Re ⁣(s)+2M1\left|\zeta\!\left(s\right) - \left(\sum_{k=1}^{N - 1} \frac{1}{{k}^{s}} + \frac{{N}^{1 - s}}{s - 1} + \frac{1}{{N}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{N}^{2 k - 1}}\right)\right)\right| \le \frac{4 \left|\left(s\right)_{2 M}\right|}{{\left(2 \pi\right)}^{2 M}} \frac{{N}^{-\left(\operatorname{Re}\!\left(s\right) + 2 M - 1\right)}}{\operatorname{Re}\!\left(s\right) + 2 M - 1}
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(121s)ζ ⁣(s)1d ⁣(n)k=0n1(1)k(d ⁣(n)d ⁣(k))(k+1)s3(1+2Im ⁣(s))(3+8)nexp ⁣(Im ⁣(s)π2)   where d ⁣(k)=ni=0k(n+i1)!4i(ni)!(2i)!\left|\left(1 - {2}^{1 - s}\right) \zeta\!\left(s\right) - \frac{1}{d\!\left(n\right)} \sum_{k=0}^{n - 1} \frac{{\left(-1\right)}^{k} \left(d\!\left(n\right) - d\!\left(k\right)\right)}{{\left(k + 1\right)}^{s}}\right| \le \frac{3 \left(1 + 2 \left|\operatorname{Im}\!\left(s\right)\right|\right)}{{\left(3 + \sqrt{8}\right)}^{n}} \exp\!\left(\frac{\left|\operatorname{Im}\!\left(s\right)\right| \pi}{2}\right)\; \text{ where } d\!\left(k\right) = n \sum_{i=0}^{k} \frac{\left(n + i - 1\right)! {4}^{i}}{\left(n - i\right)! \left(2 i\right)!}

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2019-07-15 23:42:41.550119 UTC