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Fungrim entry: 65fa9f

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

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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