# On the least common multiple of several random integers

## TL;DR

This paper investigates the asymptotic distribution of the least common multiple of several random integers, providing a probabilistic criterion for convergence and identifying the limit as an infinite product of independent prime-indexed variables.

## Contribution

It introduces a probabilistic approach to analyze the distribution of the LCM of random integers and extends existing theorems by deriving new convergence criteria and explicit limit representations.

## Key findings

- Established a criterion for distributional convergence of scaled LCMs
- Identified the limit as an infinite product of independent prime-related variables
- Computed the generating function of a related geometric sum

## Abstract

Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as $n\to\infty$ of $\frac{f(L_n(k))}{n^{rk}}$ for a wide class of multiplicative arithmetic functions~$f$ with polynomial growth $r>-1$. Furthermore, we identify the limit as an infinite product of independent random variables indexed by 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 Erd\H{o}s and Wintner (1939), Fern\'{a}ndez and Fern\'{a}ndez (2013) and Hilberdink and T\'{o}th (2016).