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

Abstract

Let $L_n$ be the least common multiple of a random set of integers obtained from $\{1,\dots ,n\}$ 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.

arXiv comment19 pages, 2 figures