A Hybrid Mean Value Involving Dedekind Sums and the General Exponential Sums

The main purpose of this paper is using the analytic method, A. Weil's classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums and give a sharp asymptotic formula for it.

In fact, the estimate (9) is the best one, since if ( ) = 2 , then from Gauss famous work (see [7, page 195 The content and form of this paper are different from the references [3,6]. Conrey et al. [3] studied the general 2 th power mean of Dedekind sums and obtained an asymptotic formula. Zhang [6] only obtained a relationship between (ℎ, ) and the mean square value of Dirichlet -functions. Our work is using Zhang's result, Weil's classical work for the upper bound estimate of the general exponential sums, and the properties of Gauss sums to study the hybrid mean value problem involving Dedekind sums and the general exponential sums ( ( ), ) and give a sharper asymptotic formula for it.

Main Theorems
In this paper, we will obtain the following two results.
where denotes the solution of the congruence equation For the classical Kloosterman sums, we can also obtain a similar conclusion. That is, we have the following.

Theorem 2.
Let be an odd prime; then we have the asymptotic formula where ( , ) = ∑

Lemmas and Proofs of the Theorems
In order to complete the proof of our theorems, we need the following several simple lemmas. Hereinafter, we will use many definitions and properties of Gauss sums, Kloosterman sums, and character sums, all of which can be found in [7,[10][11][12][13], so they will not be repeated here. First we have the following.

Lemma 3. Let be an odd prime and let be the Dirichlet character mod . Then one has the estimate
where ∑ mod (−1)=−1 denotes the summation over all odd characters mod .
Proof. From the method of proving Lemma 5 in [14] we may immediately deduce this estimate.

Lemma 4. Let > 2 be an integer; then for any integer with
( , ) = 1, one has the identity where (1, ) denotes the Dirichlet -function corresponding to character mod .
The Scientific World Journal 3 Proof. See Lemma 2 of [6].

Lemma 5. Let be an odd prime; then one has the asymptotic formula
Proof. See the theorem and corollary of [14].