Fungrim home page

Riemann hypothesis

Table of contents: Definitions - Formal statement - Statements equivalent to the Riemann hypothesis - Formulas conditional on the Riemann hypothesis - Generalizations

Definitions

c03de4
Symbol: RiemannHypothesis RH\operatorname{RH} Riemann hypothesis

Formal statement

Related topics: Riemann zeta function, Zeros of the Riemann zeta function
9fa2a1
(RH)    (for all s with sCand0Re(s)1andζ(s)=0,Re(s)=12)\left(\operatorname{RH}\right) \iff \left(\text{for all } s \text{ with } s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, 0 \le \operatorname{Re}(s) \le 1 \,\mathbin{\operatorname{and}}\, \zeta(s) = 0, \,\, \operatorname{Re}(s) = \frac{1}{2}\right)
49704a
(RH)    (for all nZ1,Re ⁣(ρn)=12)\left(\operatorname{RH}\right) \iff \left(\text{for all } n \in \mathbb{Z}_{\ge 1}, \,\, \operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}\right)

Statements equivalent to the Riemann hypothesis

Prime counting function

Related topic: Prime numbers
bfaeb5
(RH)    (for all x[2,),π(x)li(x)<xlog(x))\left(\operatorname{RH}\right) \iff \left(\text{for all } x \in \left[2, \infty\right), \,\, \left|\pi(x) - \operatorname{li}(x)\right| < \sqrt{x} \log(x)\right)

Robin's criterion

3142ec
(RH)    (for all nZ5041,σ1 ⁣(n)<eγnlog ⁣(log(n)))\left(\operatorname{RH}\right) \iff \left(\text{for all } n \in \mathbb{Z}_{\ge 5041}, \,\, \sigma_{1}\!\left(n\right) < {e}^{\gamma} n \log\!\left(\log(n)\right)\right)
e4287f
(RH)    (for all nZ2,σ1 ⁣(n)<Hn+exp ⁣(Hn)log ⁣(Hn))\left(\operatorname{RH}\right) \iff \left(\text{for all } n \in \mathbb{Z}_{\ge 2}, \,\, \sigma_{1}\!\left(n\right) < H_{n} + \exp\!\left(H_{n}\right) \log\!\left(H_{n}\right)\right)

Li's criterion

Related topic: Keiper-Li coefficients
e68f82
(RH)    (for all nZ1,λn>0)\left(\operatorname{RH}\right) \iff \left(\text{for all } n \in \mathbb{Z}_{\ge 1}, \,\, \lambda_{n} > 0\right)
a5d65f
(RH)    (n=1λna(n)2<   where a(n)=log(n)2log ⁣(2π)+1γ2)\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right)

Landau's function

Related topic: Landau's function
65fa9f
(RH)    (for all nZ1,log ⁣(g(n))<f(n)   where f(y)=solution*x(1,)[li(x)=y])\left(\operatorname{RH}\right) \iff \left(\text{for all } n \in \mathbb{Z}_{\ge 1}, \,\, \log\!\left(g(n)\right) < \sqrt{f(n)}\; \text{ where } f(y) = \mathop{\operatorname{solution*}\,}\limits_{x \in \left(1, \infty\right)} \left[\operatorname{li}(x) = y\right]\right)

Definite integrals

7783f9
(RH)    (1π0log ⁣(ζ ⁣(12+it)ζ ⁣(12))1t2dt=π8+γ4+log ⁣(8π)42)\left(\operatorname{RH}\right) \iff \left(\frac{1}{\pi} \int_{0}^{\infty} \log\!\left(\left|\frac{\zeta\!\left(\frac{1}{2} + i t\right)}{\zeta\!\left(\frac{1}{2}\right)}\right|\right) \frac{1}{{t}^{2}} \, dt = \frac{\pi}{8} + \frac{\gamma}{4} + \frac{\log\!\left(8 \pi\right)}{4} - 2\right)
cf70ce
(RH)    (0112t2(1+4t2)31/2log ⁣(ζ ⁣(σ+it))dσdt=π(3γ)32)\left(\operatorname{RH}\right) \iff \left(\int_{0}^{\infty} \frac{1 - 12 {t}^{2}}{{\left(1 + 4 {t}^{2}\right)}^{3}} \int_{1 / 2}^{\infty} \log\!\left(\left|\zeta\!\left(\sigma + i t\right)\right|\right) \, d\sigma \, dt = \frac{\pi \left(3 - \gamma\right)}{32}\right)

De Bruijn-Newman constant

22ab47
Symbol: DeBruijnNewmanLambda Λ\Lambda De Bruijn-Newman constant
a71ddd
(RH)    (Λ=0)\left(\operatorname{RH}\right) \iff \left(\Lambda = 0\right)

Formulas conditional on the Riemann hypothesis

e6ff64
Re ⁣(ρn)=12\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2}
375afe
π(x)li(x)<xlog(x)8π\left|\pi(x) - \operatorname{li}(x)\right| < \frac{\sqrt{x} \log(x)}{8 \pi}

Generalizations

Related topic: Dirichlet L-functions
dc593e
Symbol: GeneralizedRiemannHypothesis GRH\operatorname{GRH} Generalized Riemann hypothesis
e2a734
(GRH)    (for all (q,χ,n) with qZ1andχGqandnZ{0},Re ⁣(ρn,χ)=12)\left(\operatorname{GRH}\right) \iff \left(\text{for all } \left(q, \chi, n\right) \text{ with } q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \setminus \left\{0\right\}, \,\, \operatorname{Re}\!\left(\rho_{n,\chi}\right) = \frac{1}{2}\right)

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

2019-10-05 13:11:19.856591 UTC