Abstract
The main purpose of this paper is to introduce a new mixed 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 -functions with the weight of these mixed exponential sums.
1. Introduction
Let be an integer, and let be a Dirichlet character . Then, for any integer , the famous Gauss sums are defined as follows: where .
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.
From the properties of characters mod , we known that if is a primitive character mod , then , and . Many other properties of and can also be found in [1–4].
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.
Theorem 1. Let be an odd prime; let be any nonprincipal character . Then, for any integers , , and , one has the estimate
Theorem 2. Let be an odd prime; let be any nonprincipal character . Then, for any integers and , one has the identity
Theorem 3. Let be an odd prime. Then, for any integers , , and with , one has the asymptotic formula where is the principal character , denotes the greatest common divisor of , , and , and .
In Theorem 1, we only discussed the case, in which there exist two variables. For general case (with () variable), whether there exists a sharp estimate for the sums is an interesting problem.
Let ; whether there exists an exact computational formula for the th power mean, is also an open problem.
2. 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, 5–7], 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 .
Proof. If , then, from the properties of Gauss sums and quadratic residue , we have
where denotes the Legendre symbol.
Since is a nonprincipal Dirichlet character mod , from (8), the properties of Gauss sums, and complete residue system mod , we have
For any nonprincipal character , we have . So, from (9) and noting that , we have
This proves Lemma 4.
Lemma 5. Let be an odd prime; let be any nonprincipal character . Then, for any integers and , one has the estimate
Proof. Since is a nonprincipal character , if and , then
If and or and or , then, from the results of Weil [8], Malyšev [7], and Estermann [6], with some minor modifications, we can deduce the estimate
where denotes the greatest common divisor of , , and .
Now Lemma 5 follows from (12) and (13).
Lemma 6. Let be an odd prime; then, for any integer with , one has the calculating formula
Proof. See [9] or Corollary 2 of [10].
Lemma 7. Let be an odd prime and let be the Dirichlet character . Then one has the estimate
Proof. See Lemma 5 of [11].
3. Proof of the Theorems
In this section, we will complete the proof of our theorems. First we prove Theorem 1. In fact, from Lemmas 4 and 5, we may immediately deduce the estimate This proves Theorem 1.
Theorem 2 follows from Lemmas 4 and 6. In fact, from these two lemmas, we have This proves Theorem 2.
Now, we prove Theorem 3. Note that the asymptotic formula is and the estimate for Kloosterman sums (see [6]) is as follows: from Lemmas 4 and 7 and the method of proving the theorem in [11], we have This completes the proof of Theorem 3.
Conflict of Interests
The authors declare that they have no conflict of interests.
Authors’ Contribution
Zhan Yu obtained the theorems and completed the proof. Li Xiaoxue corrected and improved the final version. Both authors read and approved the final paper.
Acknowledgments
The authors would like to thank the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P. S. F. (2013JZ001) and N. S. F. (11371291) of China.