Asymptotics of Arithmetic Functions of GCD and LCM of Random Integers in Hyperbolic Regions

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