Limit theorems for the least common multiple of a random set of integers

TL;DR

This paper investigates the asymptotic behavior of the logarithm of the least common multiple of a randomly selected subset of integers from 1 to n, establishing weak convergence, laws of large numbers, and Poisson limit theorems.

Contribution

It introduces new limit theorems for the least common multiple of random subsets, including weak convergence to a Gaussian process and Poisson approximations.

Findings

Weak convergence of the normalized log L_n process to a Gaussian process

Strong law of large numbers for log L_n

Poisson limit theorems under varying regimes of θ

Abstract

Let $L_{n}$ be the least common multiple of a random set of integers obtained from $\{1,\ldots,n\}$ by retaining each element with probability $\theta\in (0,1)$ independently of the others. We prove that the process $(\log L_{\lfloor nt\rfloor})_{t\in [0,1]}$, after centering and normalization, converges weakly to a certain Gaussian process that is not Brownian motion. Further results include a strong law of large numbers for $\log L_{n}$ as well as Poisson limit theorems in regimes when $\theta$ depends on $n$ in an appropriate way.