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

Let q≥3 be an odd number,a be any fixed positive integer with(a,q)=1.For each integerb with 1≤)b<q and(b,q)=1,it is clear that there exists one and only one c with 0<c<q suchthat 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)-(1/2)φ(q).The main purpose of this paper is to study the distribution properties of E(a,q),and give a sharperhybrid mean-value formula involving E(a,q)and general Kloosterman sums.