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The main purpose of this paper is to study the calculating problem of one kind hybrid power mean involving the cubic Gauss sums and Kloosterman sums, and using the elementary methods, analytic methods and the properties of the classical Gauss sums to give some interesting calculating formula for them. At the same time, the paper also provides an effective calculating method for the study of the hybrid power mean involving the k-th Gauss sums and Kloosterman sums.

We consider the calculation problem of one kind hybrid power mean involving the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and use the analytic method and the properties of classical Gauss sums to give some identities and asymptotic formulas for them.

Abstract We apply the analytic method and the properties of the classical Gauss sums to study the computational problem of a certain hybrid power mean of the trigonometric sums and to prove several new mean value formulae for them. At the same time, we also obtain a new recurrence formula involving the Gauss sums and two-term exponential sums.

The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and acquire several accurate asymptotic formulas for them.

The main purpose of this paper is using the analytic method, the properties of trigonometric sums and Gauss sums to study the computational problem of one kind hybrid power mean involving two different trigonometric sums, and give an interesting computational formula for it.

The main purpose of this paper is to study the computational problem of one kind hybrid power mean involving two-term exponential sums and quartic Gauss sums using the analytic method and the properties of the classical Gauss sums, and to prove some interesting fourth-order linear recurrence formulae for this problem. As an application of our result, we can also obtain an exact computational formula for one kind congruence equation mod p, an odd prime.

Abstract In this paper, we use the analytic method and the properties of classical Gauss sums to study the computational problems of one kind hybrid power mean of quartic Gauss sums and two-term exponential sums, and give an interesting fourth-order linear recurrence formula for it.

The main purpose of this paper is using the analytic method and the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving two-term exponential sums and polynomial character sums. Then the authors give some interesting calculating formulae for them. [ABSTRACT FROM AUTHOR]

We apply the analytic method and the properties of the classical Gauss sums to study the computational problem of a certain hybrid power mean of the trigonometric sums and to prove several new mean value formulae for them. At the same time, we also obtain a new recurrence formula involving the Gauss sums and two-term exponential sums. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the analytic method and the properties of the character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and the two-term exponential sums, and give several interesting identities and asymptotic formulae for them. [ABSTRACT FROM AUTHOR]

Abstract The main purpose of this paper is using the analytic method and the orthogonal properties of the character sums to study the computational problem of one kind hybrid power mean of the generalized Gauss sums mod p $\operatorname{mod}\ p$ , an odd prime with p ≡ 1 mod 3 $p\equiv 1\operatorname{mod}\ 3$ , and give some exact computational formulas for this kind hybrid power mean.

The main purpose of this paper is using the analytic method and the orthogonal properties of the character sums to study the computational problem of one kind hybrid power mean of the generalized Gauss sums $\operatorname{mod}\ p$ , an odd prime with $p\equiv 1\operatorname{mod}\ 3$ , and give some exact computational formulas for this kind hybrid power mean. [ABSTRACT FROM AUTHOR]

In this paper, we find the 2k-th power mean of the inversion of L-functions with the weight of the general quartic Gauss sums. We establish the results with the help of Dirichlet characters and properties of classical Gauss sums. We also describe asymptotic behaviour for it. [ABSTRACT FROM AUTHOR]