Abstract
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.
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