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

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 Tóth’s work, where the conditions are pairwise.