Abstract
In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the properties of the error term of the fourth power mean of the generalized cubic Gauss sums and give two recurrence formulae for it.
1. Introduction
For any integer and any Dirichlet character , the definition of the classical Gauss sums iswhere is an integer and .
This sum and its properties are of great significance to the analytic number theory, and many number theory problems are closely related to them. Therefore, it is necessary to study the various properties of and related sums. In this paper, we consider the generalized -th Gauss sums , which is defined as follows:where is any positive integer and is an integer with .
It is clear that this sum is a generalization of the classical Gauss sums . In fact, . Of course, the value of is irregular as varies. However, some scholars have found that has good value distribution properties in some problems of weighted mean value, even if we can get their exact calculation formulae for some th power mean. In addition, there are some good upper bound estimates for . For example, for any integer with , from the general result of Cochrane and Zheng [1], we can deducewhere denotes the number of distinct prime divisors of . The case that is a prime is due to Weil [2].
For , by the results of Zhang [3], let be any integer with , and there are the following two identities:where denotes the Legendre symbol modulo .
Zhang and Liu [4] have studied the sumand obtained the following calculation formula:where is a prime with and is a real constant.
However, the value of was not given in [4], and the form of was not concise. Now, for any integer , we let
In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the th power mean of and give two recurrence formulae for it. That is, we shall prove the following main results.
Theorem 1. Let be an odd prime with . Then, for any positive integer and integer with , we have the third-order linear recurrence formula:where and are defined asand , where is uniquely determined by (see [5]).
Theorem 2. Let be an odd prime with . Then, for any positive integer , we have the recurrence formulawhere the first three terms of are
Theorem 3. Let be an odd prime with . Then, for any positive integer , we have the third-order linear recurrence formulawhere the initial values of are
Taking in Theorem 3, we may immediately deduce the following corollary.
Corollary 1. Let be an odd prime with ; then, we have the identity
2. Several Lemmas
In this section, we give three lemmas which are necessary in the proofs of our theorems. In the process of proving our lemmas, we need some knowledge of the analytic number theory; all of which can be found in [6–8], so it is not necessary to repeat them here.
Lemma 1. Let be an odd prime with . Then, for any third-order character , we havewhere is uniquely determined by and .
Proof. This result can be found in [9] or [10].
Lemma 2. Let be an odd prime with . Then, for any cubic character , we have the identities
Proof. For any integer , it is easy to show thatFrom the properties of the cubic character modulo , we haveSo, we have the identityTherefore,This proves Lemma 2.
Lemma 3. Let be an odd prime with . Then, for any cubic character modulo , we have the identitywhereand is defined as the same as in Lemma 2.
Proof. From Lemma 2, we haveNote that , and from (17), we haveTherefore,In addition,Using the method similar to (24), we obtainCombining formulae (23)–(27), we haveThis proves Lemma 3.
3. Proofs of the Theorems
Now, we shall complete the proofs of our main results. Firstly, we prove Theorem 1. Let be an odd prime with , be any Dirichlet character modulo , and be a cubic character modulo . Then, from the properties of the classical Gauss sums and (17), we havewhere is the same as in Lemma 2.
For any integer , we have the trigonometric identity
From (30), we have the identity
For any positive integer , from Lemma 3, we havewhere
This proves Theorem 1.
Now, we prove Theorem 2. From Theorem 1, we have
From Lemma 2, we have
From (36) and Lemma 3, we have
Now, Theorem 2 follows from (35)–(38).
Finally, we prove Theorem 3. For any integer , from Lemma 3, we have
Therefore, we have
Combining (39)–(43), we may immediately complete the proof of Theorem 3.
4. Conclusion
The main results of this paper give two third-order linear recurrence formulae for the error term of the fourth power mean of the generalized cubic Gauss sums, and these results are the improvement and generalization of [4]. They are some new contributions in the relevant fields.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are 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 funded by the N. S. F. (11771351) of P. R. China and Xi’an Science and Technology Plan Innovation Fund “Special Project for Xi'an University” 2020KJWL08.