On the Fourth Power Mean of the Two-Term Exponential Sums

Abstract

The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of two-term exponential sums and give an interesting identity and asymptotic formula for it.

1. Introduction

Let be a positive integer. For any integers and , the two-term exponential sum is defined as follows: where .

Regarding the various properties of , some authors have studied them, and obtained a series of results. Some related works can be found in references [1–8]. For example, Gauss’ classical work proved the remarkable formula (see [1]) where .

Generally, for any odd number and , the exact value of is , More relevant to it, Cochrane and Zheng [4] showed the general sum that where denotes the number of all distinct prime divisors of .

In this paper, we study the fourth power mean of the two-term exponential sum as follows: where is any integer with .

Regarding this problem, it seems that none has studied it yet; at least we have not seen any related result before. The problem is interesting, because it can reflect that the mean value of is well behaved. The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the special case of (4) with , , an odd prime and give an interesting identity and asymptotic formula for it. That is, we will prove the following conclusion.

Theorem 1. Let be a prime. Then for any integer with , one has the identity where denotes the Legendre symbol.
For any prime with , one cannot give an exact computational formula in our theorem at present. The difficulty is that one needs to know the value of the character sums where is any -order character .
For any integer , whether there exists an exact computational formula for is an open problem, where is an odd prime and .

2. Several Lemmas

In this section, we will give several lemmas which are necessary in the proof of our theorem. In the proving process of all lemmas, we used many properties of Gauss sums; all these can be found in [1], so they will not be repeated here. First we have the following.

Lemma 2. Letting be an odd prime with , then one has the identity where denotes the Legendre's symbol.

Proof. For any prime , note that if passes through a complete residue system , then also passes through a complete residue system , so note the identity (this formula can be found in Hua’s book, Section 7.8, Theorem 8.2 [9]). One has This proves Lemma 2.

Lemma 3. Let be an odd prime be any nonprincipal character . Then for any integer with , one has the identity

Proof. Note that is a non-principal character , so if is not a -order character , (i.e., , the principal character ), then from the properties of Gauss sums we have where denotes the classical Gauss sums.
If is a -order character , then ; note that for any integer with , we have , , , and From the method of proving (12) we have the identity Now note that if . From (12) and (14) we may immediately deduce Lemma 3.

Lemma 4. Let be an odd prime and let be a th character . Then one has the identity Therefore

Proof. Noting that , from the definition and properties of the classical Gauss sums, we have or Similarly, we also have or This proves Lemma 4.

3. Proof of the Theorem

In this section, we shall complete the proof of our theorem. First from the orthogonality of characters we have On the other hand, if , then any non-principal character is not a -order character . Note that From (22) and Lemma 3 we have If , then combining (21) and (23) we may immediately deduce the identity or If , since is a 5-order character , and are also 5-order characters , then note that From Lemma 4 and the method of proving (23) we have So if , then combining (21) and (27) we can deduce the asymptotic formula Now our theorem follows from (25) and (28).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N. S. F. (11371291), S. R. F. D. P. (20136101110014), and N. S. F. (2013JZ001) of Shaanxi Province of China.