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

2025/05/08 by Chan Ieong Kuan, Kuan, Chan Ieong
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical functions and polynomials #Number Theory (math.NT) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2505.05630

openalex publication_date 2025/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Tóth's work, where the conditions are pairwise.

Citations

• Pairwise Relative Primality of Positive Integers
• Pairwise non-coprimality of triples

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