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 (. 1, . 2, c; q) (0 < . 1, . 2 . 1) the number of all pairs of integers a, b satisfying ab . c(mod q), 1 . a . . 1 q, 1 . b . . 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 (. 1, . 2, c; q), and give a sharp asymptotic formula for it.