On a Problem of D. H. Lehmer and 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 $E(a, q)=N(a, q)-{1\over 2}\phi (q)$. The main purpose of this paper is to study the distribution properties of E(a, q), and to give a sharper hybrid mean value formula involving E(a, q) and Kloosterman sums.