Fungrim home page

Landau's function

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

Definitions

32e430
Symbol: LandauG g(n)g(n) Landau's function
6af603
g(n)=A000793 ⁣(n)g(n) = \text{A000793}\!\left(n\right)

Tables

177218
Table of g(n)g(n) for 0n1000 \le n \le 100

Arithmetic representations

7932c3
g(n)=max{lcm ⁣(s1,,sk):kZ0andsiZ1andi=1ksi=n}g(n) = \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

a3ab2a
limnlog ⁣(g(n))nlog(n)=1\lim_{n \to \infty} \frac{\log\!\left(g(n)\right)}{\sqrt{n \log(n)}} = 1

Bounds and inequalities

9697b8
log ⁣(g(n))nlog(n)(1+log ⁣(log(n))0.9752log(n))\log\!\left(g(n)\right) \le \sqrt{n \log(n)} \left(1 + \frac{\log\!\left(\log(n)\right) - 0.975}{2 \log(n)}\right)
3d5019
log ⁣(g(n))nlog(n)\log\!\left(g(n)\right) \ge \sqrt{n \log(n)}
87d19b
max{p:pPandpg(n)}1.328nlog(n)\max \left\{ p : p \in \mathbb{P} \,\mathbin{\operatorname{and}}\, p \mid g(n) \right\} \le 1.328 \sqrt{n \log(n)}

Riemann hypothesis

65fa9f
(RH)    (for all 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 \in \mathbb{Z}_{\ge 1}, \,\, \log\!\left(g(n)\right) < \sqrt{f(n)}\; \text{ where } f(y) = \mathop{\operatorname{solution*}\,}\limits_{x \in \left(1, \infty\right)} \left[\operatorname{li}(x) = y\right]\right)

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

2019-11-19 15:10:20.037976 UTC