Abstract
The goal of this paper is to solve the computational problem of one kind rational polynomials of classical Gauss sums, applying the analytic means and the properties of the character sums. Finally, we will calculate a meaningful recursive formula for it.
Keywords:
third-order character; classical Gauss sums; rational polynomials; analytic method; recursive formula 2010 Mathematics Subject Classification:
11L05; 11L07
1. Introduction
Let be an integer. For any Dirichlet character , according to the definition of classical Gauss sums , we can write
where .
Since this sum appears in numerous classical number theory problems, and it has a close connection with the trigonometric sums, we believe that classical Gauss sums play a crucial part in analytic number theory. Because of this phenomenon, plenty of experts have researched Gauss sums. Meanwhile, more conclusions have been obtained as regards their arithmetic properties. Such as the following results provided by Chen and Zhang []:
Let p be an odd prime with , be any fourth-order character . Then one has the identity
where denotes the the Legendre’s symbol (please see Reference [,] for its definition and related properties), and .
If p is a prime with , is any third-order character , then Zhang and Hu [] had already obtained an analogous result (see Lemma 1). However, perhaps the most beautiful and important property of Gauss sums is that , for any primitive character .
Reference [] and References [,,,,,,,,,] have a good deal of various elementary properties of Gauss sums. In this paper, the following rational polynomials of Gauss sums attract our attention.
where p is an odd prime, k is a non-negative integer, is any non-principal character .
Observing the basic properties of Equation (1), we noticed that hardly anyone had published research in any academic papers to date. We consider that the question is significant. In addition, the regularity of the value distribution of classical Gauss sums could be better revealed. Presently, we will explain certain properties discovered in our investigation. See that has some good properties. In fact, for some special character , the second-order linear recurrence formula for for all integers may be found similarly.
The goal of this paper is to use the analytic method and the properties of the character sums to solve the computational problem of , and to calculate two recursive formulae, which are listed hereafter:
Theorem 1.
Let p be a prime with , ψ be any third-order character . Then, for any positive integer k, we can deduce the following second-order linear recursive formulae
where the initial values and , d is uniquely determined by and .
So we can deduce the general term
Theorem 2.
Let p be a prime with , ψ be any third-order character . Then, for any positive integer k, we will obtain the second-order linear recursive formulae
where the initial values , and .
Similarly, we can also deduce the general term
2. Several Lemmas
We have used five simple and necessary lemmas to prove our theorems. Hereafter, we will apply relevant properties of classical Gauss sums and the third-order character , all of which can be found in books concerning elementary and analytic number theory, such as in References [,], so we will not duplicate the related contents.
Lemma 1.
If p is any prime with , ψ is any third-order character , then, we have the equation
where denotes the classical Gauss sums, d is uniquely determined by and .
Proof.
See References [] or []. □
Lemma 2.
Let p be a prime with , ψ be any third-order character , denotes the Legendre’s symbol . The following identity holds
Proof.
Firstly, using the properties of Gauss sums, we get
On the other side, we get the sums
Now, Lemma 2 has been proved. □
Lemma 3.
Let p be a prime with , χ be any sixth-order character . Then, about classical Gauss sums , the following holds:
where , d is uniquely determined by and .
Proof.
Since , is a third-order character . Any sixth-order character can be denoted as or . Note that , and , from Lemma 2 we deduce
and
Note that is a real character , , and . If ; , , if . From Equation (6) we may immediately prove the sum
Let , then is a sixth-order character and . From Equation (7) we can deduce the sum term
The proof of Lemma 3 has been completed. □
Lemma 4.
Let p be a prime with , ψ be any three-order character . Then, we compute the sum term
Proof.
Let be a three-order character . Then, for any six-order character , we must have or . Without loss of generality we suppose that , then note that , and Theorem 7.5.4 in Reference [], we acquire
Using the properties of Gauss sums we can write
Noting that , we can deduce
Similarly, we can see
This completes the proof of Lemma 4. □
Lemma 5.
Let p be a prime with , ψ be any three-order character . Then, we obtain the sum term
Proof.
From Lemma 3 and the method of proving Lemma 4 we can easily deduce Lemma 5. □
3. Proofs of the Theorems
In this section, we prove our two theorems. For Theorem 1, since , is a third-order character , then for any positive integer k, let
From Lemma 5 we have
and
Combining Equations (12) and (13) we may immediately compute the second-order linear recursive formula
with initial values and .
Note that the two roots of the equation are
So from Equation (14) and its initial values we may immediately deduce the general term
where . Now Theorem 1 has been finished.
Similarly, from Lemma 4 and the method of proving Theorem 1 we can easily obtain Theorem 2. Now, we have completed all the proofs of our Theorems.
4. Conclusions
The main results of this paper are Theorem 1 and 2. They give a new second-order linear recurrence formula for Equation (1) with the third-order character . Therefore, we can calculate the exact value of Equation (1). Note that , so is a unit root, thus, the results in this paper profoundly reveal the distributional properties of two different Gauss sums quotients on the unit circle.
For the other characters, for example, the fifth-order character with , we naturally ask whether there exists a similar formula as presented in our theorems. This is still an open problem. It will be the content of our future investigations.
Funding
This research was funded by [National Natural Science Foundation of China] Grant number [11771351].
Acknowledgments
The author wish to express her gratitude to the editors and the reviewers for their helpful comments.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Chen, Z.Y.; Zhang, W.P. On the fourth-order linear recurrence formula related to classical Gauss sums. Open Math. 2017, 15, 1251–1255. [Google Scholar]
- Apostol, T.M. Introduction to Analytic Number Theory; Springer: New York, NY, USA, 1976. [Google Scholar]
- Zhang, W.P.; Hu, J.Y. The number of solutions of the diagonal cubic congruence equation mod p. Math. Rep. 2018, 20, 73–80. [Google Scholar]
- Chen, L.; Hu, J.Y. A linear recurrence formula involving cubic Gauss sums and Kloosterman sums. Acta Math. Sin. 2018, 61, 67–72. [Google Scholar]
- Li, X.X.; Hu, J.Y. The hybrid power mean quartic Gauss sums and Kloosterman sums. Open Math. 2017, 15, 151–156. [Google Scholar]
- Zhang, H.; Zhang, W.P. The fourth power mean of two-term exponential sums and its application. Math. Rep. 2017, 19, 75–83. [Google Scholar]
- Zhang, W.P.; Liu, H.N. On the general Gauss sums and their fourth power mean. Osaka J. Math. 2005, 42, 189–199. [Google Scholar]
- Berndt, B.C.; Evans, R.J. The determination of Gauss sums. Bull. Am. Math. Soc. 1981, 5, 107–128. [Google Scholar] [CrossRef]
- Berndt, B.C.; Evans, R.J. Sums of Gauss, Jacobi, and Jacobsthal. J. Number Theory 1979, 11, 349–389. [Google Scholar] [CrossRef]
- Hua, L.K. Introduction to Number Theory; Science Press: Beijing, China, 1979. [Google Scholar]
- Kim, T. Power series and asymptotic series associated with the q analog of the two-variable p-adic L-function. Russ. J. Math. Phys. 2005, 12, 186–196. [Google Scholar]
- Kim, H.S.; Kim, T. On certain values of p-adic q-L-function. Rep. Fac. Sci. Eng. Saga Univ. Math. 1995, 23, 1–2. [Google Scholar]
- Chae, H.; Kim, D.S. L function of some exponential sums of finite classical groups. Math. Ann. 2003, 326, 479–487. [Google Scholar] [CrossRef]
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