Fungrim home page

Fungrim entry: e4287f

(RH)    (σ1 ⁣(n)<Hn+exp ⁣(Hn)log ⁣(Hn)   for all nZ2)\left(\operatorname{RH}\right) \iff \left(\sigma_{1}\!\left(n\right) < H_{n} + \exp\!\left(H_{n}\right) \log\!\left(H_{n}\right) \;\text{ for all } n \in \mathbb{Z}_{\ge 2}\right)
References:
  • https://doi.org/10.2307/2695443
TeX:
\left(\operatorname{RH}\right) \iff \left(\sigma_{1}\!\left(n\right) < H_{n} + \exp\!\left(H_{n}\right) \log\!\left(H_{n}\right) \;\text{ for all } n \in \mathbb{Z}_{\ge 2}\right)
Definitions:
Fungrim symbol Notation Short description
RiemannHypothesisRH\operatorname{RH} Riemann hypothesis
DivisorSigmaσk ⁣(n)\sigma_{k}\!\left(n\right) Sum of divisors function
Expez{e}^{z} Exponential function
Loglog(z)\log(z) Natural logarithm
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("e4287f"),
    Formula(Equivalent(RiemannHypothesis, All(Less(DivisorSigma(1, n), Add(HarmonicNumber(n), Mul(Exp(HarmonicNumber(n)), Log(HarmonicNumber(n))))), ForElement(n, ZZGreaterEqual(2))))),
    References("https://doi.org/10.2307/2695443"))

Topics using this entry

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

2020-08-27 09:56:25.682319 UTC