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 $\varphi(n)$ be the Euler totient function. We also give an upper bound on the error term in an asymptotic formula for $\sum_{n=1}^H (\varphi(n)/n)^m$ for $m \ge 2$ and as $H \rightarrow \infty$.