Counting tuples restricted by coprimality conditions

Abstract

Given a set $A=\{(i_1,j_1),\ldots,(i_m,j_m)\}$ we say that $(a_1,\ldots,a_v)$ exhibits pairwise coprimality if $\gcd(a_i,a_j) = 1$ for all $(i,j)\in A$. For a given positive $x$ we give an asymptotic formula for the number of $(a_1,\ldots,a_v)$ with $1 \le a_1,\ldots,a_v \le x$ that exhibit pairwise coprimality. Our error term is better than that of Hu.