Abstract
The main purpose of this article is using the elementary methods and the properties of the quadratic residue modulo an odd prime to study the calculating problem of the fourth power mean of one kind two-term exponential sums and give an interesting calculating formula for it.
1. Introduction
Let be a fixed integer. For any integer and integer with , we define the two-term exponential sums as follows:where, as usual, and denotes the imaginary unit, that is .
Since this kind of sums play a very important role in the study of analytic number theory, so many number theorists and scholars had studied the various properties of and obtained a series of meaningful research results, we do not want to enumerate here, and interested readers can refer to [1–16]. For example, Zhang and Zhang [1] proved that for any odd prime , one haswhere represents any integer with .
Shen and Zhang [2] obtained an interesting recurrence formula forwhere is an odd prime with .
Chen and Zhang [3] proved that for any prime with , one has the identitywhere , denotes Legendre’s symbol modulo , and .
Zhang and Han [4] used the elementary method to obtain the identitywhere denotes an odd prime with .
Chen and Wang [5] studied the calculating problem of the fourth power mean of and proved the following conclusion.
Let be an odd prime, then one has the identitywhere and denotes Legendre’s symbol modulo .
Zhang and Zhang [6] proved that for any prime , one has the identitywhere is an integer which satisfies the estimate .
Liu and Zhang [7] proved that for any prime with , one has the identitywhere denotes the summation over all Dirichlet characters, modulo .
Inspired by the works in [5, 6], in this paper, we consider the following calculating problem of the -th power mean of the two-term exponential sums:where is an odd prime and is an integer.
About this problem, it seems that none had studied it before; at least we have not seen such a result at present. In this paper, we will use the properties of the solutions of the congruence equations and the quadratic residue to study this problem and give an interesting calculating formula for (9) with . That is, we will prove the following result.
Theorem 1. Let be an odd prime, then we have the identitywhere and denotes Legendre’s symbol modulo .
Note that the estimate (see [17] for general results), and from this theorem and [5], we may immediately deduce the following several corollaries.
Corollary 1. Let be an odd prime with , then we have the asymptotic formula
Corollary 2. Let be an odd prime with , then we have the asymptotic formula
Some notes: the constant in our theorem has a special meaning. In fact, for any prime with , one has the identity (see [18])where is any quadratic nonresidue modulo . That is, .
For any prime and integer , we naturally ask whether there is an exact calculating formula for (9)?
This is an open problem. We believe this to be true. We even have the following.
Conjecture 1. Let be an odd prime with . Then, for any integer , there are two integers and depending only on and , such that the identity
2. Several Lemmas
In this section, we will give several necessary lemmas. Of course, the proofs of some lemmas need the knowledge of elementary and analytic number theory. In particular, the properties of the quadratic residues and the Legendres symbol modulo are going to be used. All these can be found in [15, 18–20], and we do not repeat them. First, we have the following lemma.
Lemma 1. Let be an odd prime with . Then, for any fourth-order character , we have the identitywhere denotes the classical Gauss sums, , and denotes Legendre’s symbol modulo .
Proof. See Lemma 2 in Chen and Zhang [3].
Lemma 2. Let be an odd prime with , then we have the identitywhere is defined as in Lemma 1.
Proof. For any odd prime with , let be any fourth-order character modulo , andThen, from the properties of the classical Gauss sums, we havewhere . Note that , , from (18) and Lemma 1, we havewhere . If , then and , so we haveIf , then and , so we haveNow, if , then from (20) and Lemma 1, we haveIf , then from (21) and Lemma 1, we haveCombining (22) and (23) and , we may immediately deduce Lemma 2.
Lemma 3. Let be an odd prime with . Then, we have the identity
Proof. From the methods of proving Lemma 2, we haveIf , then note that , and from (18), we haveThen, applying (25), we haveIf , then , and from (19) and (25), we haveNow, Lemma 3 follows from (27) and (28).
Lemma 4. Let be a prime with . Then, we have the identity
Proof. From the properties of the complete residue system modulo , we haveNote that the identityand , if ; , if . We haveFrom (30) and (32), we haveThis proves Lemma 4.
Lemma 5. Let be a prime with . Then, we have the identity
Proof. It is clear thatFrom (31), we haveNote that and , and from the properties of complete residue system modulo and , we haveCombining (35)–(37) and Lemma 3, we haveThis proves Lemma 5.
3. Proof of the Theorem
Applying several basic lemmas in Section 2, we can easily complete the proof of our theorem. In fact, for any odd prime , if , then from the properties of the reduced residue system modulo , we have
So, in this case, we have the identity
If , then we have
If , then note that , and from (41), Lemma 2, Lemma 4, and Lemma 5, we may immediately deduce
Similarly, if , then note that , and from (41), Lemma 2, Lemma 4, and Lemma 5, we may immediately deduce
Now, the theorem follows from (40), (42), and (43).
This completes the proof of our main result.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the NSF (11862018) of People’s Republic of China and NSF (2017BS0101) of Inner Mongolia of China.