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

Abstract

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 $$ E{\left( {a,q} \right)} = N{\left( {a,q} \right)} - \frac{1} {2}\phi {\left( q \right)} $$ . 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.