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Riemann zeta function

Table of contents: Definitions - Illustrations - Dirichlet series - Euler product - Laurent series - 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

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

Euler product

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ζ ⁣(s)=∏p11−1ps\zeta\!\left(s\right) = \prod_{p} \frac{1}{1 - \frac{1}{{p}^{s}}}

Laurent series

Related topic: Stieltjes constants
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ζ ⁣(s)=1s−1+∑n=0∞(−1)nn!γn(s−1)n\zeta\!\left(s\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n} {\left(s - 1\right)}^{n}

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 1≤n≤201 \le n \le 20
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Table of ζ ⁣(−n)\zeta\!\left(-n\right) for 0≤n≤300 \le n \le 30
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Table of ζ ⁣(n)\zeta\!\left(n\right) to 50 digits for 2≤n≤502 \le n \le 50

Analytic properties

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ζ ⁣(s) is holomorphic on s∈C∖{1}\zeta\!\left(s\right) \text{ is holomorphic on } s \in \mathbb{C} \setminus \left\{1\right\}
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poles⁡ s∈C∪{∞~}ζ ⁣(s)={1}\mathop{\operatorname{poles}\,}\limits_{s \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \zeta\!\left(s\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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Symbol: RiemannZetaZero — ρn\rho_{n} — Nontrivial zero of the Riemann zeta function
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Symbol: RiemannHypothesis — RH⁡\operatorname{RH} — Riemann hypothesis
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(RH⁡)  ⟺  (Re⁡ ⁣(ρn)=12   for all n∈Z≥1)\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right)
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zeros⁡ s∈Rζ ⁣(s)={−2n:n∈Z≥1}\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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zeros⁡ s∈C, 0≤Re⁡(s)≤1ζ ⁣(s)={ρn:n∈Z  and⁡  n≠0}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 \le \operatorname{Re}(s) \le 1} \zeta\!\left(s\right) = \left\{ \rho_{n} : n \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; n \ne 0 \right\}
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zeros⁡ s∈Cζ ⁣(s)={−2n:n∈Z≥1}∪{ρn:n∈Z  and⁡  n≠0}\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 < \operatorname{Re}\!\left(\rho_{n}\right) < 1
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Re⁡ ⁣(ρn)=12\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}
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ρ−n=ρn‾\rho_{-n} = \overline{\rho_{n}}
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Table of Im⁡ ⁣(ρn)\operatorname{Im}\!\left(\rho_{n}\right) to 50 digits for 1≤n≤501 \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)s−1sin⁡ ⁣(πs2)Γ ⁣(1−s)ζ ⁣(1−s)\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)
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ζ ⁣(1−s)=2cos⁡ ⁣(12πs)(2π)sΓ(s)ζ ⁣(s)\zeta\!\left(1 - s\right) = \frac{2 \cos\!\left(\frac{1}{2} \pi s\right)}{{\left(2 \pi\right)}^{s}} \Gamma(s) \zeta\!\left(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}(s)\right)
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∣ζ ⁣(s)∣<3∣1+s1−s∣∣1+s2π∣(1+η−Re⁡(s))/2ζ ⁣(1+η)\left|\zeta\!\left(s\right)\right| < 3 \left|\frac{1 + s}{1 - s}\right| {\left|\frac{1 + s}{2 \pi}\right|}^{\left( 1 + \eta - \operatorname{Re}(s) \right) / 2} \zeta\!\left(1 + \eta\right)

Euler-Maclaurin formula

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ζ ⁣(s)=∑k=1N−11ks+N1−ss−1+1Ns(12+∑k=1MB2k(2k)!(s)2k−1N2k−1)−∫N∞B2M ⁣(t−⌊t⌋)(2M)!(s)2Mts+2M dt\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=1N−11ks+N1−ss−1+1Ns(12+∑k=1MB2k(2k)!(s)2k−1N2k−1))∣≤4∣(s)2M∣(2π)2MN−(Re⁡(s)+2M−1)Re⁡(s)+2M−1\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}(s) + 2 M - 1\right)}}{\operatorname{Re}(s) + 2 M - 1}
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∣(1−21−s)ζ ⁣(s)−1d(n)∑k=0n−1(−1)k(d(n)−d(k))(k+1)s∣≤3(1+2∣Im⁡(s)∣)(3+8)ne∣Im⁡(s)∣π/2   where d(k)=n∑i=0k(n+i−1)!4i(n−i)!(2i)!\left|\left(1 - {2}^{1 - s}\right) \zeta\!\left(s\right) - \frac{1}{d(n)} \sum_{k=0}^{n - 1} \frac{{\left(-1\right)}^{k} \left(d(n) - d(k)\right)}{{\left(k + 1\right)}^{s}}\right| \le \frac{3 \left(1 + 2 \left|\operatorname{Im}(s)\right|\right)}{{\left(3 + \sqrt{8}\right)}^{n}} {e}^{\left|\operatorname{Im}(s)\right| \pi / 2}\; \text{ where } d(k) = n \sum_{i=0}^{k} \frac{\left(n + i - 1\right)! {4}^{i}}{\left(n - i\right)! \left(2 i\right)!}

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC