On the generalization of the D. H. Lehmer problem

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. Supported by National Natural Science Foundation (Grant No. 10601039) of China