Pairwise non-coprimality of triples

Abstract

We say that $(a_1,...,a_k)$ is pairwise non-coprime if $\gcd (a_i,a_j)\ne 1$ for all $1\le i<j\le k$. Let $a_1,a_2,a_3$ be positive integers less than $H$. We obtain an asymptotic formula for the number of $(a_1,a_2,a_3)$ that are pairwise non-coprime. The probability that a randomly chosen unbounded positive integer triple is pairwise non-coprime is approximately 17.4%. Let $\phi (n)$ be the Euler totient function. We also give an upper bound on the error term in an asymptotic formula for ${\sum }_{n=1}^H(\phi (n)/n{)}^m$ for $m\ge 2$ and as $H\to \infty$.

arXiv comment8 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

References in corpus (1)

• On the Number of Eisenstein Polynomials of Bounded Height

Cited by in corpus (1)

• Errata and Addenda to Mathematical Constants