Central limit theorem for the least common multiple of a uniformly sampled m-tuple of integers

Abstract

Let Bn(m) be a set picked uniformly at random among all m-elements subsets of {1,2,…,n}. We provide a pathwise construction of the collection (Bn(m))1⩽m⩽n and prove that the logarithm of the least common multiple of the integers in (Bn(⌊mt⌋))t⩾0, properly centered and normalized, converges to a Brownian motion when both m,n tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of m independent random variables having uniform distribution on {1,2,…,n}. Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.