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).

The main purpose of this paper is using a mean value theorem of Dirieh-let L-ihnctions to study the asymptotic property of a hybrid mean value of a Coehrane sum, and givc an intercsting mean 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 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.

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≥3 is an odd number, a is any fixed positlvc integer with (a, q)=1. For each integer b with 1≤b<q and (b, q)=1, it is clear that there exists one and only one c with 0<c<q such that bc≡a (rood q). Lct N (a, q) denotcs thc numbcr of all solutions of the congruent equation bc≡a (mod q) for 1≤b, c<q in which b and c arc of opposite parity, and lct E(a, q)=N(a, q)-Ф(q). The main purpose of this paper is to study the distribution properties of E(a,q), and give a sharper hybrid mean value formula involving E (a, q) and Kloostcrman sums.