THE DISTRIBUTION OF RELATIVELY r-PRIME INTEGERS IN RESIDUE CLASSES

If 1 is the only r-th power which is a divisor of m₁, m₂, . . . , mk, then m₁, m₂, . . . , mk, are said to be relatively r-prime. If ā = (a₁, a₂,..., ak) is a k-tuple of nonnegative integers, h is a positive integer and x is a positive real number, let Q(x; ā, h, r, k) denote the number of k-tuples of positive integers (m₁, m₂, . . . , mk) for which 1 ≤ mi ≤ x, mi ≡ ai (mod h), i = 1,2 , . . . , k and m₁, m₂ , . . . , mk are relatively r-prime. An asymptotic formula with 0-estimate for Q(x; ā, h, r, k) is determined. Special cases of this estimate give earlier estimates for relatively prime integers and r-free integers.