Multivariate multiplicative functions of uniform random vectors in large integer domains

Abstract

For a wide class of sequences of integer domains ${\mathcal{D}}_n\subset {\mathbb{N}}^d$, $n\in \mathbb{N}$, we prove distributional limit theorems for $F(X_1^{(n)},\dots ,X_d^{(n)})$, where $F$ is a multivariate multiplicative function and $(X_1^{(n)},\dots ,X_d^{(n)})$ is a random vector with uniform distribution on ${\mathcal{D}}_n$. As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set $\{X_1^{(n)},\dots ,X_d^{(n)}\}$. This generalizes previously known limit results for ${\mathcal{D}}_n$ being either a discrete cube or a discrete hyperbolic region.

21 pages