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Zeros of the Riemann zeta function

Table of contents: Main properties - Numerical values - Related topics

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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
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Symbol: RiemannHypothesis — RH⁡\operatorname{RH} — Riemann hypothesis

Main properties

See also: Riemann hypothesis
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(RH⁡)  ⟺  (Re⁡(s)=12   for all s∈C with 0≤Re⁡(s)≤1  and⁡  ζ ⁣(s)=0)\left(\operatorname{RH}\right) \iff \left(\operatorname{Re}(s) = \frac{1}{2} \;\text{ for all } s \in \mathbb{C} \text{ with } 0 \le \operatorname{Re}(s) \le 1 \;\mathbin{\operatorname{and}}\; \zeta\!\left(s\right) = 0\right)
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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}}

Numerical values

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ρ1∈12+[14.134725141734693790457251983562470270784257115699±2.44⋅10−49]i\rho_{1} \in \frac{1}{2} + \left[14.134725141734693790457251983562470270784257115699 \pm 2.44 \cdot 10^{-49}\right] i
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ρ2∈12+[21.022039638771554992628479593896902777334340524903±2.19⋅10−49]i\rho_{2} \in \frac{1}{2} + \left[21.022039638771554992628479593896902777334340524903 \pm 2.19 \cdot 10^{-49}\right] i
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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
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Table of Im⁡ ⁣(ρn)\operatorname{Im}\!\left(\rho_{n}\right) to 10 digits for 1≤n≤5001 \le n \le 500
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Table of Im⁡ ⁣(ρ10n)\operatorname{Im}\!\left(\rho_{{10}^{n}}\right) to 50 digits for 0≤n≤160 \le n \le 16

Related topics: Riemann zeta function

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2021-03-15 19:12:00.328586 UTC