Landau’s function for one million billions

Abstract

Let (formula presented) denote the symmetric group with n letters, and g(n) the maximal order of an element of (formula presented) . If the standard factorization of M into primes is (formula presented), we define ℓ(M) to be (formula presented) one century ago, E. Landau proved that (formula presented) and that, when n goes to infinity, (Formula Presented). There exists a basic algorithm to compute g(n) for 1≤n≤N; its running time is O (formula presented) and the needed memory is O(N); it allows computing g(n) up to, say, one million. We describe an algorithm to calculate g(n) for n up to 1015. The main idea is to use the so-called ℓ-superchampion numbers. Similar numbers, the superior highly composite numbers, were introduced by S. Ramanujan to study large values of the divisor function τ(n)=∑d|n1.