Multivariate multiplicative functions of uniform random vectors in large integer domains

TL;DR

This paper establishes distributional limit theorems for multivariate multiplicative functions of uniform random vectors in large integer domains, generalizing previous results for specific geometric regions.

Contribution

It introduces a general framework for limit theorems of multiplicative functions over broad classes of integer domains, extending prior specific cases.

Findings

Limit theorems for GCD and LCM of random vectors

Generalization beyond hypercube and hyperbolic regions

Unified approach for various 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 $F(X_1^{(n)},\ldots,X_d^{(n)})$, where $F$ is a multivariate multiplicative function and $(X_1^{(n)},\ldots,X_d^{(n)})$ 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 $\{X_1^{(n)},\ldots,X_d^{(n)}\}$. This generalizes previously known limit results for $\mathcal{D}_n$ being either a discrete cube or a discrete hyperbolic region.