On the Asymptotic Density of $k$-tuples of Positive Integers Satisfying Arbitrary GCD Conditions

TL;DR

This paper develops asymptotic formulas for counting k-tuples of positive integers satisfying arbitrary gcd conditions, generalizing previous pairwise results using multivariable Dirichlet series.

Contribution

It introduces a general framework for counting k-tuples under arbitrary gcd constraints, extending prior pairwise condition analyses.

Findings

Established existence conditions for such k-tuples.

Derived asymptotic formulas using multivariable Dirichlet series.

Generalized previous pairwise gcd condition results.

Abstract

We consider the problem of counting $k$-tuples of positive integers satisfying any arbitrary set of gcd conditions, where every integer is not larger than $x$. We first establish the conditions to guarantee the existence of such tuples, and then obtain asymptotic formulae for the count of such tuples with the help of a multivariable Dirichlet series. Part of this work can be viewed as a generalization of T\'oth's work, where the conditions are pairwise.