Abstract
The main purpose of this paper is to study the computational problem of one kind rational polynomials of the classical Gauss sums, and using the purely algebraic methods and the properties of the character sums (a prime with ) to give an exact evaluation formula for it.
Keywords:
Twelfth-order character mod p; classical Gauss sums; rational polynomials; analytic method; evaluation formula MSC:
11L05; 1L07
1. Introduction
As usual, let be an integer, be any Dirichlet character . Then the classical Gauss sum is defined as follows:
where .
This sum occupies a very vital position in the research of analytic number theory, and plenty of famous number theory problems are closely related to it. Because of this reason, many number theory experts have studied the properties of the classical Gauss sums, and obtained a series of important conclusions. For example, Z. Y. Chen and W. P. Zhang [] provided the following result:
Let p be an odd prime with , be any fourth-order character . Then one has the identity
where indicates the the Legendre symbol , and denotes for convenience.
H. Bai and J. Y. Hu [] used identity (1) to obtain a second-order linear recursive formula for one kind rational polynomials involving the classical Gauss sums. That is, let p be a prime with , be any eighth-order character , and
Then we have the second-order linear recursive formula
where , , and is the same as in (1).
W. P. Zhang and J. Y. Hu [] (the different form can also be found in B. C. Berndt and R. J. Evans [] ) proved that for any prime p with and any third-order character , one has the equation
where denotes the classical Gauss sums, d is uniquely determined by and .
Chen Li [] used the identity (2) to prove the following conclusions. Let
Then for any prime p with and any third-order character , one has the second-order linear recursive formula
where the initial values and , d is the same as in (1).
Therefore, from (3) one can deduce the general term
If p be a prime with , then one has
where the initial values , .
Similarly, from (4) one can also deduce the general term
Other works related to the classical Gauss sums and trigonometric sums can also be found in the references [,,,,,,,,,] . Here we will not list them one by one.
In this paper, as a note of [,], we will study a similar problem for prime p is a prime with and any twelfth-order character . More specifically, let be any twelfth-order character and
Then we can use the purely algebraic methods and the properties of the classical Gauss sums to give a evaluation formula for (5). That is, we have the following:
Theorem 1.
Let p be a prime with , χ be any twelfth-order character . Then for any positive integer k, we have the second-order linear recursive formula
where the initial values and , d is uniquely determined by and .
From this recursive formula we may immediately deduce the general term
where .
2. Two Simple Lemmas
In this part, two necessary lemmas in the proof process of our theorem will be given. Hereafter, we will need use many properties of the classical Gauss sums, the third-order character and the fourth-order character , all of which can be found reference [], so we will not repeat them here. First we have the following:
Lemma 1.
Let p be a prime with , χ be any sixth-order character . Then about the classical Gauss sums , the following identity holds,
where , d is uniquely determined by and .
Proof.
In fact, this is Lemma 3 of [], so its proof is omitted. □
Lemma 2.
Let p be a prime with . Then for any twelfth-order character , we obtain the identity
where d is the same as in (2).
Proof.
Let be any third-order character , be any fourth-order character . Then for any twelfth-order character , we must have , , or . Without loss of generality we suppose that , then for any integer b with , note that we have the identity (see Theorem 7.5.4 in [])
where denotes the Legendre’s symbol . □
From the properties of Gauss sums we have
where we have used the identities and .
On the other hand, note that , we can also deduce that
Combining (6) and (7) we can deduce that
or
Similarly, we also have
Note that , is a sixth-order character and , from (8), (9) and Lemma 1 we may instantly deduce the identity
It is clear that if , then , and . Now our result follows from (10). This proves Lemma 2.
3. Proof of the Theorem
In this section, we complete the proof of our theorem. In fact for any prime p with and any twelfth-order character , from the definition of and Lemma 2 we have and . If integer , then we have
The formula (11) implies that
with initial values and .
This completes the proof of our theorem.
4. Conclusions
The main purpose of this paper is to use the properties of the classical Gauss sums to give an interesting second-order linear recurrence formula for one kind special Gauss sums, so we obtained an exact evaluation formula for the general term of this kind sums. Of course, the sums in our paper is quite different from the sums in references [,]. This provides a new way of thinking, direction and method for us to further study on more general problems.
It is obvious that if p has a twelfth-order character , then p just has different twelfth-order characters , , and , they all appear in (5), and conjugated configuration.
For any prime p with , it also just has different fifth-order characters , we define as follows:
Then we think there must be a positive integer r so that has a second-order linear recurrence formula as in our theorem. In fact, for a certain positive integer r, we only need to know .
This still is an open problem. We will continue to study.
Author Contributions
All authors have equally contributed to this work. All authors read and approved the final manuscript.
Funding
This work is supported by Hainan Provincial N. S. F. (118MS041) and the N. S. F. (11771351, 11501452) of P. R. China.
Acknowledgments
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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