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

Abstract

Copyright © 2018, arXiv, All rights reserved. Let Ln be the least common multiple of a random set of integers obtained from f1; : : : ; ng by retaining each element with probability θ ϵ (0, 1) independently of the others. We prove that the process (log L⌊nt⌋)tϵ[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 Ln as well as Poisson limit theorems in regimes when θ depends on n in an appropriate way.60F05 (Primary) 11N37, 60F15 (secondary)