Fungrim entry: e4287f

\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
`RiemannHypothesis`	$\operatorname{RH}$	Riemann hypothesis
`DivisorSigma`	$\sigma_{k}\!\left(n\right)$	Sum of divisors function
`Exp`	${e}^{z}$	Exponential function
`Log`	$\log(z)$	Natural logarithm
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	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

Riemann hypothesis