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

Chan Ieong Kuan

TL;DR

The paper tackles counting $k$-tuples $(n_1,\dots,n_k)$ with $n_i\le x$ that satisfy an arbitrary set of gcd-conditions. It develops a multivariable Dirichlet-series framework, decomposing conditions into prime components and applying Möbius convolution to extract the main term, with precise local factors shaping the Euler-product constant. A key admissibility criterion is established: $(\mathcal{Q},f)$ is admissible if and only if for every prime $p$ and every $T\in \mathcal{Q}$, $g_p(T)=\min\{v_i^{(p)}: i\in T\}$. For admissible sets, the count has the asymptotic form $A_{\mathcal{G}} x^k + O(x^{k-1}(\log x)^{k-1})$, where $A_{\mathcal{G}}$ is an explicit Euler product, generalizing known constants from Tóth’s results to arbitrary gcd-condition configurations. This provides a unified analytic toolkit for gcd-constrained tuple counts with potential broad applications in analytic number theory.

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.

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

TL;DR

The paper tackles counting -tuples with that satisfy an arbitrary set of gcd-conditions. It develops a multivariable Dirichlet-series framework, decomposing conditions into prime components and applying Möbius convolution to extract the main term, with precise local factors shaping the Euler-product constant. A key admissibility criterion is established: is admissible if and only if for every prime and every , . For admissible sets, the count has the asymptotic form , where is an explicit Euler product, generalizing known constants from Tóth’s results to arbitrary gcd-condition configurations. This provides a unified analytic toolkit for gcd-constrained tuple counts with potential broad applications in analytic number theory.

Abstract

We consider the problem of counting -tuples of positive integers satisfying any arbitrary set of gcd conditions, where every integer is not larger than . 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.
Paper Structure (8 sections, 3 theorems, 43 equations)

This paper contains 8 sections, 3 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Main results
  4. Admissibility of a set of gcd conditions
  5. Asysmptotic density satisfying a specified set of gcd conditions
  6. The region of convergence for $D'(s_1,\dots,s_k)$
  7. The primes that show up in the gcd conditions
  8. Proof of Theorem \ref{['AsymThm']}

Key Result

Theorem 3.1

A set of gcd conditions $(\mathcal{Q},f)$ is admissible if and only if for any prime $p$, any element $T \in \mathcal{Q}$, $g_p(T) = \operatorname{min} \{ v_i^{(p)} | i \in T \}$.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 3.1
  • Theorem 3.2
  • ...and 4 more