A problem of D.H. Lehmer and its mean square value formula

Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1≤b<q, there is a unique integer c coprime to q with 1≤c<q such that bc ≡ a(mod q). Let N(a, q) denote the number of solutions of the congruence equation bc ≡ a(mod q) with 1≤b, c<q such that b, c are of opposite parity. The main purpose of this paper is to use the properties of Dedekind sums, the properties of Cochrane sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the mean square value ∑′_a=1^q (N(a, q) - 1/2 φ(q))², and give a sharp asymptotic formula.