The distribution of relatively \(r\)-prime integers in residue classes (Q1802710)

The natural numbers \(m_ 1,m_ 2,\dots,m_ k\) are called relatively \(r\)-prime, if no \(r\)-th power of an integer greater than 1 is a common divisor of all \(k\) numbers. An asymptotic representation is proved for the number of \(k\)-tuples of positive integers \((m_ 1,m_ 2,\dots,m_ k)\) for which \(1\leq m_ i\leq x\), \(m_ i\equiv a_ i\pmod h\), \(i=1,2,\dots,k\), and \(m_ 1,m_ 2,\dots,m_ k\) are relatively \(r\)- prime. The asymptotic formula is rather general and contains as special cases some known estimates for relatively prime integers and \(r\)-free numbers.