On a Problem of D. H. Lehmer and General Kloosterman Sums

Let q 3 be an odd number, a be any fixed positive integer with (a, q) = 1. For each integer b with 1 b > q and (b, q) = 1, it is clear that there exists one and only one c with 0 > c > q such that bc a (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bc a (mod q) for 1 b, c > q in which b and c are of opposite parity, and let E(a,q)=N(a,q)12(q). The main purpose of this paper is to study the distribution properties of E(a, q), and give a sharper hybrid mean-value formula involving E(a, q) and general Kloosterman sums.