Abstract
The main purpose of this study is to use the elementary and analytic methods and the properties of the classical Gauss sums to study the calculation problems of one kind of hybrid power mean involving the quadratic character sums and the two-term exponential sums and give an exact computational formula for it.
1. Introduction
Let , , and be three positive integers with and . For any integers and , the two-term exponential sums is defined as follows:where and .
The study of the properties of the two-term exponential sum plays an important role in analytic number theory, and many famous number theory problems such as Waring’s problem and Goldbach’s conjecture are closely related to it. Because of this, many scholars have studied various properties of the two-term exponential sums and obtained a series of valuable research results [1–5].
In addition, for any odd prime , the quadratic character modulo is called Legendre’s symbol, and its definition is
It is the introduction of this symbol that greatly facilitates the study of quadratic residue properties and promotes the development of analytic number theory. Some related important works can be seen in [6–15]. For example, let and be two distinct odd primes; then, one has the quadratic reciprocal formula (see [16]: Theorem 9.8 or [17]: Theorems 4–6).
For any odd prime with , there must be two nonzero integers and such that (see [17]: Theorems 4–11)where is any quadratic nonresidue modulo and is the inverse of . That is, satisfies the equation .
Now, for any odd prime with and integer with , we define quadratic character sums as follows:
Then, for any integer , we consider the -th hybrid power mean
In this paper, we tried to give an exact computational formula for (6). That is, we will prove the following several results.
Theorem 1. Let be a prime with . If 2 is a cubic residue modulo , then we have the identity
If 2 is not a cubic residue modulo , then we have the identitywhere is uniquely determined by and .
Theorem 2. Let be a prime with . If 2 is a cubic residue modulo , then we have the identity
Theorem 3. Let be a prime with . If 2 is a cubic residue modulo , then we have the identity
If 2 is not a cubic residue modulo , then we have the identity
Theorem 4. Let be a prime with . Then, for any integer , we have the third-order linear recursive formula
Note that 2 is a cubic residue modulo if and only if is an even number (see Corollary 1 in [18]). So from Theorems 1, 2, and 4, we may immediately deduce the following two corollaries.
Corollary 1. Let be a prime with and . If is an even number, then we have the identity
Corollary 2. Let be a prime with and . If is an even number, then we have the identityNotes: in Theorems 1–3, why do we want to distinguish whether 2 is a cubic residue modulo ? The reason is that we need to determine the exact value of , where is a third-order character modulo . If 2 is a cubic residue modulo , then ; if 2 is not a cubic residue modulo , then .
In Theorem 2, we only discussed the case and 2 is a cubic residue modulo . In fact, if 2 is not a cubic residue modulo , then we can also give an exact computational formula for it, but in this case, the result looks very cumbersome and not beautiful. Corollaries 1 and 2 are the same things. Hence, all these results are not listed.
In addition, in this paper, we did not consider prime with , because in this case, the result is trivial. In fact in this case, if passes through a complete residue system modulo , then also passes through a complete residue system modulo . Hence, we have the identity
2. Some Lemmas
In this section, we will give several simple lemmas and they are all necessary in the proofs of our theorems. In addition, we also need some properties of the classical Gauss sums and character sums; all these can be found in [16, 17, 19], and we will not repeat them here. First we have the following.
Lemma 1. Let be a prime with . Then, for any third-order character modulo , we have the identitywhere denotes the classical Gauss sums, , , and is the same as defined the theorem.
Lemma 2. Let be an odd prime. Then, for any nonprincipal character modulo , we have the identitywhere denotes Legendre’s symbol modulo .
Proof. In fact, this is the product formula of Gauss sums. We can see the general results in [20, 22, 23]. Related works can also be found in [24, 25].
Lemma 3. Let be an odd prime. Then, for any integer with and any third-order character modulo , we have the identity
Proof. denotes Legendre’s symbol modulo . Then, from the properties of the third-order character modulo and classical Gauss sums, we haveNote that the identityTaking in Lemma 2, we haveNote that ; from (20) and (21), we haveSimilarly, we also haveNote that ; combining (19), (22), and (23), we have the identityThis proves Lemma 3.
Lemma 4. Let be any odd prime with ; then, for a three-order character modulo , we have the identity
Proof. For any integer with , note that the identityand from the properties of the Gauss sums and (21), we haveThis proves Lemma 4.
Lemma 5. Let be any odd prime with ; then, for any integers and with , we have the third-order linear recursive formulawhere is the same as defined in Theorem 1.
Proof. Note that ; as the principal character modulo , from Lemmas 1 and 3, we haveSo, for any integer , from (29), we have the third-order recursive formula as follows:This proves Lemma 5.
Lemma 6. Let be any odd prime with ; then, we have
Proof. From the properties of the trigonometrical sums, we haveIf , then note that the congruence equation has four solutions ; from (32), we have the identityIf , note that the congruence equation has two solutions ; so from (32), we haveNow, Lemma 6 follows from formulae (33) and (34).
3. Proofs of the Theorems
In this section, we will complete the proofs of our theorems. First we prove Theorem 1. Note that ; from Lemmas 1, 3, and 4, we have
If 2 is a cubic residue modulo , then from (35), we have
If 2 is not a cubic residue modulo , then note that the identities and , from (35), we have
Now, Theorem 1 follows from identities (36) and (37).
Now, we prove Theorem 2. From Lemma 3, we have
From (38) and Lemma 4, we have
If 2 is a cubic residue modulo , then from (39) and Lemmas 1 and 6, we have
This proves Theorem 2.
Now, we prove Theorem 3. If 2 is a cubic residue modulo , then from Theorem 1 and Lemma 6, we have
Similarly, if 2 is not a cubic residue modulo , then from Theorem 1 and Lemma 6, we also have
Now, Theorem 3 follows from (41) and (42).
Theorem 4 follows from Lemma 5.
From Theorems 1, 2, and 4 and Lemmas 5 and 6, we may immediately deduce Corollaries 1 and 2.
This completes the proofs of our all results.
4. Conclusion
The main results of this study give four theorems for the hybrid power mean involving the quadratic character sums and the two-term exponential sums. In addition, we also obtained a third-order linear recursive formula for (6). That is, for any integer , we have the recursive formulawhere is uniquely determined by and .
So, for any integer , we have solved the calculation problem of (6) completely.
Data Availability
The proof of our results is given in the paper, and no other additional data were used.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors have equally contributed to this work. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the N. S. F. (12126357) of P. R. China.