Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions

TL;DR

This paper establishes limit theorems for GCD and LCM of random integers uniformly distributed in hyperbolic regions, extending classical results to new geometric domains and providing asymptotic estimates for integer points.

Contribution

It introduces limit theorems for GCD and LCM in hyperbolic regions, a novel setting compared to classical hypercube distributions, and derives asymptotic point counts.

Findings

Limit theorems for GCD and LCM in hyperbolic regions

Asymptotic estimates for integer points in hyperbolic domains

Extension of classical results to new geometric settings

Abstract

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.