Landau's function for one million billions

Abstract

Let ${\mathfrak{S}}_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of ${\mathfrak{S}}_n$. If the standard factorization of $M$ into primes is $M=q_1^{\al_1}q_2^{\al_2}... q_k^{\al_k}$, we define $\ell (M)$ to be $q_1^{\al_1}+q_2^{\al_2}+... +q_k^{\al_k}$; one century ago, E. Landau proved that $g(n)={\max }_{\ell (M)\le n}M$ and that, when $n$ goes to infinity, $\log g(n)\sim \sqrt{n\log (n)}$. There exists a basic algorithm to compute $g(n)$ for $1\le n\le N$; its running time is $\co(N^{3/2}/\sqrt{\log N})$ and the needed memory is $\co(N)$; it allows computing $g(n)$ up to, say, one million. We describe an algorithm to calculate $g(n)$ for $n$ up to $10^{15}$. The main idea is to use the so-called {\it $\ell$-superchampion numbers}. Similar numbers, the {\it superior highly composite numbers}, were introduced by S. Ramanujan to study large values of the divisor function $τ(n)=\sum_{d\dv n} 1$.