A Brownian weak limit for the least common multiple of a random m-tuple of integers

Abstract

Let $B_n(m)$ be a set picked uniformly at random among all $m$-elements subsets of $\{1,2,\dots ,n\}$. We provide a pathwise construction of the collection $(B_n(m){)}_{1\le m\le n}$ and prove that the logarithm of the least common multiple of the integers in $(B_n(\lfloor mt\rfloor ){)}_{t\ge 0}$, 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 $\{1,2,\dots ,n\}$. 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.

34 pages

References in corpus (3)

• On the Amount of Dependence in the Prime Factorization of a Uniform Random Integer
• On the probability distribution of the gcd and lcm of $r$-tuples of integers
• On the least common multiple of several random integers