Multivariate Multiplicative Functions of Uniform Random Vectors in Large Integer Domains

For a wide class of sequences of integer domains D n ⊂ N d , n ∈ N , we prove distributional limit theorems for F ( X 1 ( n ) , … , X d ( n ) ) , where F is a multivariate multiplicative function and ( X 1 ( n ) , … , X d ( n ) ) is a random vector with uniform distribution on D n . As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set { X 1 ( n ) , … , X d ( n ) } . This generalizes previously known limit results for D n being either a discrete cube or a discrete hyperbolic region.