# Fungrim entry: 65fa9f

$\left(\operatorname{RH}\right) \iff \left(\log\!\left(g(n)\right) < \sqrt{f(n)} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\; \text{ where } f(y) = \mathop{\operatorname{solution*}\,}\limits_{x \in \left(1, \infty\right)} \left[\operatorname{li}(x) = y\right]\right)$
References:
• Marc Deleglise, Jean-Louis Nicolas, The Landau function and the Riemann Hypothesis, https://arxiv.org/abs/1907.07664
TeX:
\left(\operatorname{RH}\right) \iff \left(\log\!\left(g(n)\right) < \sqrt{f(n)} \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\; \text{ where } f(y) = \mathop{\operatorname{solution*}\,}\limits_{x \in \left(1, \infty\right)} \left[\operatorname{li}(x) = y\right]\right)
Definitions:
Fungrim symbol Notation Short description
RiemannHypothesis$\operatorname{RH}$ Riemann hypothesis
Log$\log(z)$ Natural logarithm
LandauG$g(n)$ Landau's function
Sqrt$\sqrt{z}$ Principal square root
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
UniqueSolution$\mathop{\operatorname{solution*}\,}\limits_{x \in S} Q(x)$ Unique solution
LogIntegral$\operatorname{li}(z)$ Logarithmic integral
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
Source code for this entry:
Entry(ID("65fa9f"),
Formula(Equivalent(RiemannHypothesis, Where(All(Less(Log(LandauG(n)), Sqrt(f(n))), ForElement(n, ZZGreaterEqual(1))), Equal(f(y), UniqueSolution(Brackets(Equal(LogIntegral(x), y)), ForElement(x, OpenInterval(1, Infinity))))))),
References("Marc Deleglise, Jean-Louis Nicolas, The Landau function and the Riemann Hypothesis, https://arxiv.org/abs/1907.07664"))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC