The main purpose of this paper is to study the asymptotic property of the fburth power mean of the general Kloostermaxl sums, and give an interesting calculating 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 e coprimc to q with 1≤e≤q such that be≡a (rood q). Let N (a, q) denote the number of solutions of the congruence equation bc≡a (rood q) wlth 1≤b, c<q such that b,c are of opposite parity. The main purpose of this paper is using the properties of Dcdekind sums, the properties of Cochranc sums and the mean value theorem of Diriehlet L-thnetions to study the asymptotic property of the mean square value Σ(N(a,q)-2-1(q)2, and give a sharp asymptotic formula.

In this paper, we use the properties of Gauss sums, primitive characters and the mean value theorems of Dirlehlet L-functions to study the hybrid mean value of Cochrane suiils and general Kloosterman suins and give two sharp asymptotic formulac.

The main purpose of this paper is using the mean value theorem of Dirichlet L-fhnetions to study the asymptotic property of one class number-theoretic ihnctions, and give an interesting mean square value formula.

Let p>3 bc a prime, k bc any fixed positlvc integer. For any polynomiM f(x, y) with rational integer coefficients, we define set A=A(f,k,δ)={(a,b):f(a,b)=O (modp), {ak-p}-{bk-p}<δ} whcrc 0<δ≤1, [x] denotes the grcatest intcger not exceeding x and {x}=x-[x]. The main purpose of this paper is to study the asymptotic behaviour of M(p,A,K)=Σpa=1 Σpb=1(a,b)4 and give an interesting asymptotic formula for M (p,A,k) for some special polynomials f (x, y).