Counting tuples restricted by coprimality conditions

TL;DR

This paper derives an asymptotic formula for counting v-tuples within a range that satisfy specific pairwise coprimality conditions, improving the error term over previous results.

Contribution

It provides a new asymptotic formula with a better error term for counting tuples exhibiting pairwise coprimality under given restrictions.

Findings

Asymptotic formula for the count of coprime tuples

Improved error term compared to previous work

Enhanced understanding of coprimality restrictions in tuple counting

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.