New Mixed Exponential Sums and Their Application

Themain purpose of this paper is to introduce a newmixed exponential sums and then use the analytic methods and the properties of Gauss sums to study the computational problems of the mean value involving these sums and give an interesting computational formula and a sharp upper bound estimate for these mixed exponential sums. As an application, we give a new asymptotic formula for the fourth power mean of Dirichlet L-functions with the weight of these mixed exponential sums.


Introduction
Let  ≥ 3 be an integer, and let  be a Dirichlet character mod .Then, for any integer , the famous Gauss sums (, ) are defined as follows: where () =  2 .This sum and the other exponential sums (such as Kloosterman sums) play very important role in the study of analytic number theory, and many famous number theoretic problems are closely related to it.For example, the distribution of primes, Goldbach problem, the estimate of character sums, and the properties of Dirichlet -functions are some good examples.
In this paper, we introduce new mixed exponential sums as follows: (, , , ; ) where , , and  are any integers.We will study the arithmetical properties of (, , , ; ).About this problem, it seems that none has studied it yet; at least we have not seen any related results before.The problem is interesting, because this sum has a close relationship with the general Kloosterman sums, and it is also analogous to famous Gauss sums, so it must have many properties similar to these sums.It can also help us to further understand and study Kloosterman sums and Gauss sums.
The main purpose of this paper is using the analytic method and the properties of Gauss sums to study the fourth power mean of (, , , ; ) and its upper bound estimate and prove the following three conclusions.
where  0 is the principal character mod , (, , ) denotes the greatest common divisor of , , and , and exp() =   .
In Theorem 1, we only discussed the case, in which there exist two variables.For general case (with (≥ 3) variable), whether there exists a sharp estimate for the sums is an interesting problem.
Let  ≥ 3; whether there exists an exact computational formula for the 2th power mean, is also an open problem.

Several Lemmas
In this section, we will give several lemmas, which are necessary in the proof of our theorems.Hereinafter, we will use many properties of character sums, Kloosterman sums, and Gauss sums; all of these can be found in [1,[5][6][7], so they will not be repeated here.First, we have the following.

Lemma 4.
Let  be an odd prime; then, for any integers , , and , one has the identity where  denotes the solution of the congruence equation  ⋅  ≡ 1 mod .

Proof of the Theorems
In this section, we will complete the proof of our theorems.First we prove Theorem and the estimate for Kloosterman sums (see [6]) is as follows: −1 Let  be an odd prime.Then, for any integers , , and  with (,  2 +  2 − , ) ̸ = , one has the asymptotic formula
1.In fact, from Lemmas 4 and 5, we may immediately deduce the estimate           , if  is the Legendre symbol mod ;  4 (2 − 7) , if  is a complex character mod . 