# Landau's function

## Definitions

Symbol: LandauG $g(n)$ Landau's function
$g(n) = \text{A000793}\!\left(n\right)$

## Tables

Table of $g(n)$ for $0 \le n \le 100$

## Arithmetic representations

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

## Asymptotics

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

## Bounds and inequalities

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

## Riemann hypothesis

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

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