Abstract
Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c be two integers with (n, q) = (c, q) = 1. We denote by r n (δ 1, δ 2, c; q) (0 < δ 1, δ 2 ≤ 1) the number of all pairs of integers a, b satisfying ab ≡ c(mod q), 1 ≤ a ≤ δ 1 q, 1 ≤ b ≤ δ 2 q, (a, q) = (b, q) = 1 and n†(a + b). The main purpose of this paper is to study the asymptotic properties of r n (δ 1, δ 2, c; q), and give a sharp asymptotic formula for it.
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