Pairwise non-coprimality of triples

2013/09/22 by Randell Heyman, Heyman, Randell · 2 citations
Mathematics · #Advanced Topology and Set Theory #Analytic Number Theory Research #Mathematics and Applications #math.NT

paper · pdf · doi:10.48550/arxiv.1309.5578

8 pages. An anonymous referee has pointed out that the result Lemma 2 is already known. A comment to that effect has been added. It has also pointed out that the probability that three positive integers are pairwise non-coprime is known and a comment to that effect has been added. Two minor typographical errors have been corrected

arxiv created 2014/05/07 · arxiv updated 2014/05/08

Abstract

We say that (a₁,...,a_k) is pairwise non-coprime if gcd(a_i,a_j) ≠ 1 for all 1 ≤ i <j ≤ k. Let a₁,a₂,a₃ be positive integers less than H. We obtain an asymptotic formula for the number of (a₁,a₂,a₃) that are pairwise non-coprime. The probability that a randomly chosen unbounded positive integer triple is pairwise non-coprime is approximately 17.4%. Let φ(n) be the Euler totient function. We also give an upper bound on the error term in an asymptotic formula for ∑_n=1^H (φ(n)/n)^m for m ≥ 2 and as H → ∞.

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• On the Asymptotic Density of k-tuples of Positive Integers Satisfying Arbitrary GCD Conditions
• Errata and Addenda to Mathematical Constants

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