Abstract
In this paper, we introduce a new Gauss sum, and then we use the elementary and analytic methods to study its various properties and prove several interesting three-order linear recursion formulae for it.
1. Introduction
Let be an integer. For any Dirichlet character modulo , the classical Gauss sums is defined as follows:where is any integer, , and .
For convenience, we write . The Gauss sum plays a very important role in the study of elementary number theory and analytic number theory, and many number theory problems are closely related to it. Because of this, many scholars have studied its various properties and obtained a series of important results. For example, if , then we have the identity (see [1, 2])
If is any primitive character modulo , then one has also and the identity .
In addition, Zhang and Hu [3] (or Berndt and Evans [4]) studied the properties of some special Gauss sums and obtained the following interesting results: let be a prime with . Then for any third-order character modulo , one has the identitywhere is uniquely determined by and .
Chen and Zhang [5] studied the case of the fourth-order character modulo and obtained the following conclusion: let be a prime with . Then for any four-order character modulo , we have the identitywhere denotes Legendre’s symbol modulo .
The constant in (4) has a special meaning. In fact, we have the identity (For this, see Theorems 4–11 in [6])where is any quadratic nonresidue modulo . That is, .
Some other results related to various Gauss sums and their recursion properties can also be found in references [7–10], and we will not list them all here.
In this paper, we introduce a new Gauss sum , which is defined as follows: let be an odd prime. For any integer with , we definewhere denotes the Legendre’s symbol modulo .
It is clear that if , then note that ; from the properties of the reduced residue system modulo , we have
So this time, becomes the classical Gauss sum.
If , then we only knew that is a real number, if ; and is a pure imaginary number, if . In fact if , then note that , and this time we have
If , then note that , and this time we have
But beyond these relatively simple properties, we do not know anything else. In this paper, we shall focus on the calculating problems of . We shall use the analytic methods to give an interesting three-order linear recursion formula for . That is, we shall prove the following two results.
Theorem 1. Let be an odd prime with . Then for any integer , we have the recursion formulawhere is uniquely determined by and , and the three initial values , , and .
Theorem 2. Let be an odd prime with . Then for any integer , we have the recursion formulawhere is the same as in Theorem 1, and the three initial values , , and .
Of course, our theorems are also true for all integers . In particular, we have the following conclusions:
Theorem 3. For any prime with , we have the identitieswhere is the same as defined in (3), i.e., .
2. Several Lemmas
In this section, we first give several simple lemmas. Of course, the proofs of these lemmas and theorems need some knowledge of character sums and analytic number theory. They can be found in many number theory books, such as [1, 2, 6], here we do not need to list.
Lemma 1. Let be a prime with . Then for any six-order character , we have the identitywhere , is uniquely determined by , and .
Proof. For this, refer the study of Chen [11].
Lemma 2. Let be a prime with , denote Legendre’s symbol modulo , and denote any three-order Dirichlet character modulo . Then for any integer with , we have the identities
Proof. It is clear that for any integer with , from the properties of the three-order character modulo , we haveIt is clear that is a real character modulo ; from (15) and the properties of the classical Gauss sums, we haveNote that , , , , , and the identity ; from (16), we also haveFrom (16) and (17) and Lemma 1, we haveNow Lemma 2 follows from (16)–(18).
Lemma 3. Let be an odd prime with . Then for any with , we have the identities
Proof. Note that modulo 4, , , , and ; from (16) and the methods of proving Lemma 2, we also haveIt is clear that Lemma 3 follows from (20)–(22).
3. Proofs of the Theorems
Now we shall complete the proofs of our all results. First we prove Theorem 1. Let be an odd prime with , and then note that ; from Lemma 2 and the properties of the character sums modulo , we have
From (23) and Lemmas 1 and 2, we have
If , then , from Lemma 2, we have
From (23)–(26) and the definition of , we may immediately deduce the three-order linear recursion formulawith the three initial values , , and
This proves Theorem 1.
Now we prove Theorem 2. If be an odd prime with , then note that and ; from Lemma 3, we have
It is clear that from Lemmas 1 and 3, we have
From (30)–(32) and the definition of , we have the three-order linear recursion formulawhere the three initial values , , and .
This proves Theorem 2.
Now we prove Theorem 3. First for any integer with , we have . In fact if ; then from (25), we have
This is impossible. So we have .
On the contrary, if , then from (25), we have
Combining (23), (24), and (35), we haveor
From (23), (24), and (37), we also haveor
Similarly, if , then from (31), (35), Lemma 3, and the methods of proving (39), we can also deduce that
If , then is a real number, so this time we have
If , then is a pure imaginary number; this time we also have
It is clear that from (24), (32), (41), and (42), we can deduce the identity
From (39)–(42) and noting that , we can also deduce
This completes the proofs of our all results.
4. Conclusion
The main result of this paper is to prove a three-order linear recursion formula for one kind new Gauss sums. As an application of this result, we obtained following conclusion: for any prime with , we have the identities
These results not only gave the exact values for the fourth power mean and its inverse fourth power mean of a new Gauss sums, they are also some new contribution to research in related fields.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
The author contributed to the work and read and approved the final manuscript.
Acknowledgments
This work was supported by the N. S. F. (11771351) of P. R. China.