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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

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(RH)    (for all n with nZ1,Re ⁣(ρn)=12)\left(\operatorname{RH}\right) \iff \left(\text{for all } n \text{ with } n \in \mathbb{Z}_{\ge 1}, \operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}\right)
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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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zerossC,0Re(s)1ζ ⁣(s)={ρn:nZandn0}\mathop{\operatorname{zeros}\,}\limits_{s \in \mathbb{C},\,0 \le \operatorname{Re}\left(s\right) \le 1} \zeta\!\left(s\right) = \left\{ \rho_{n} : n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \ne 0 \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 < \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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ρn=ρn\rho_{-n} = \overline{\rho_{n}}

Numerical values

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ρ112+[14.134725141734693790457251983562470270784257115699±2.441049]i\rho_{1} \in \frac{1}{2} + \left[14.134725141734693790457251983562470270784257115699 \pm 2.44 \cdot 10^{-49}\right] i
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ρ212+[21.022039638771554992628479593896902777334340524903±2.191049]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 1n501 \le n \le 50
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Table of Im ⁣(ρn)\operatorname{Im}\!\left(\rho_{n}\right) to 10 digits for 1n5001 \le n \le 500
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Table of Im ⁣(ρ10n)\operatorname{Im}\!\left(\rho_{{10}^{n}}\right) to 50 digits for 0n160 \le n \le 16

Related topics: Riemann zeta function

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

2019-08-21 11:44:15.926409 UTC