On Lehmer’s problem and Dedekind sums

Abstract

Let p be an odd prime and c a fixed integer with (c, p) = 1. For each integer a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one b with 0 ⩽ b ⩽ p − 1 such that ab ≡ c (mod p). Let N(c, p) denote the number of all solutions of the congruence equation ab ≡ c (mod p) for 1 ⩽ a, b ⩽ p−1 in which a and $\overline b$ are of opposite parity, where $\overline b$ is defined by the congruence equation b$\overline b$ ≡ 1 (mod p). The main purpose of this paper is to use the properties of Dedekind sums and the mean value theorem for Dirichlet L-functions to study the hybrid mean value problem involving N(c, p)−½φ(p) and the Dedekind sums S(c, p), and to establish a sharp asymptotic formula for it.