On the least common multiple of several random integers

Abstract

Let Ln(k) denote the least common multiple of k independent random integers uniformly chosen in {1,2,…,n}. In this article, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as n→∞ of f(Ln(k))nrk for a wide class of multiplicative arithmetic functions f with polynomial growth r∈R. Furthermore, we identify the limit as an infinite product of independent random variables indexed by the set of prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016).