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Landau's function

Table of contents: Definitions - Tables - Arithmetic representations - Asymptotics - Bounds and inequalities - Riemann hypothesis

Definitions

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Symbol: LandauG g ⁣(n)g\!\left(n\right) Landau's function

Tables

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Table of g ⁣(n)g\!\left(n\right) for 0n1000 \le n \le 100

Arithmetic representations

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g ⁣(n)=max{lcm ⁣(s1,,sk):kZ0andsiZ1andi=1ksi=n}g\!\left(n\right) = \max \left\{ \operatorname{lcm}\!\left({s}_{1}, \ldots, {s}_{k}\right) : k \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, {s}_{i} \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \sum_{i=1}^{k} {s}_{i} = n \right\}

Asymptotics

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limnlog ⁣(g ⁣(n))nlog ⁣(n)=1\lim_{n \to \infty} \frac{\log\!\left(g\!\left(n\right)\right)}{\sqrt{n \log\!\left(n\right)}} = 1

Bounds and inequalities

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log ⁣(g ⁣(n))nlog ⁣(n)(1+log ⁣(log ⁣(n))0.9752log ⁣(n))\log\!\left(g\!\left(n\right)\right) \le \sqrt{n \log\!\left(n\right)} \left(1 + \frac{\log\!\left(\log\!\left(n\right)\right) - 0.975}{2 \log\!\left(n\right)}\right)
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log ⁣(g ⁣(n))nlog ⁣(n)\log\!\left(g\!\left(n\right)\right) \ge \sqrt{n \log\!\left(n\right)}
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max{p:pPandpg ⁣(n)}1.328nlog ⁣(n)\max \left\{ p : p \in \mathbb{P} \,\mathbin{\operatorname{and}}\, p \mid g\!\left(n\right) \right\} \le 1.328 \sqrt{n \log\!\left(n\right)}

Riemann hypothesis

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(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)

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2019-08-21 11:44:15.926409 UTC