# Landau's function

## Definitions

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

## Tables

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

## Arithmetic representations

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

## Asymptotics

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

## Bounds and inequalities

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

## Riemann hypothesis

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

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