Divisibility properties of polynomial expressions of random integers

Abstract

We study divisibility properties of a set {f1(Un(s)),…,fm(Un(s))}, where f1,…,fm are polynomials in s variables over Z and Un(s) is a point picked uniformly at random from the set {1,…,n}s. We show that, as n→∞, the GCD and the suitably normalized LCM of this set converge in distribution to a.s. finite random variables under mild assumptions on f1,…,fm. Our approach is based on the known fact that the uniform distribution on {1,…,n} converges to the Haar measure on the ring Zˆ of profinite integers, combined with the Lang–Weil bounds and tools from probability theory.