A New Gauss Sum and Its Recursion Properties

. In this paper


Introduction
Let q > 1 be an integer. For any Dirichlet character χ modulo q, the classical Gauss sums G(m, χ; q) is defined as follows: where m is any integer, e(y) � e 2πiy , and i 2 � − 1.
For convenience, we write τ(χ) � G (1, χ; q). e 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 (m, q) � 1, then we have the identity (see [1,2]) If χ is any primitive character modulo q, then one has also G(m, χ; q) � χ(m)τ(χ) and the identity |τ(χ)| � � q √ .
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 p be a prime with p ≡ 1 mod 3.
Chen and Zhang [5] studied the case of the fourth-order character modulo p and obtained the following conclusion: let p be a prime with p ≡ 1 mod 4. en for any four-order character χ 4 modulo p, we have the identity where ( * /p) � χ 2 denotes Legendre's symbol modulo p. e constant α � α(p) in (4) has a special meaning. In fact, we have the identity (For this, see eorems 4-11 in [6]) where r is any quadratic nonresidue modulo p. at is, Some other results related to various Gauss sums and their recursion properties can also be found in references [7][8][9][10], and we will not list them all here.
In this paper, we introduce a new Gauss sum A(m) � A(m, p), which is defined as follows: let p be an odd prime. For any integer m with (m, p) � 1, we define It is clear that if (p − 1, 3) � 1, then note that χ 3 2 � χ 2 ; from the properties of the reduced residue system modulo p, we have So this time, A(m) � G(m, χ 2 ; p) � χ 2 (m)τ(χ 2 ) becomes the classical Gauss sum.
If p ≡ 1 mod 3, then we only knew that A(m) is a real number, if p ≡ 1 mod 12; and A(m) is a pure imaginary number, if p ≡ 7 mod 12. In fact if p ≡ 1 mod 12, then note that χ 2 (− 1) � 1, and this time we have If p ≡ 7 mod 12, then note that χ 2 (− 1) � − 1, 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 G n (p). We shall use the analytic methods to give an interesting three-order linear recursion formula for G n (p). at is, we shall prove the following two results. Theorem 1. Let p be an odd prime with p ≡ 7 mod 12. en for any integer n ≥ 3, we have the recursion formula where d is uniquely determined by 4p � d 2 + 27b 2 and d ≡ 1 mod 3, and the three initial values G 0 (p) � p − 1,

Theorem 2.
Let p be an odd prime with p ≡ 1 mod 12. en for any integer n ≥ 3, we have the recursion formula where d is the same as in eorem 1, and the three initial Of course, our theorems are also true for all integers n < 0. In particular, we have the following conclusions: where b is the same as defined in (3), i.e., 4p � d 2 + 27b 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. ey can be found in many number theory books, such as [1, 2, 6], here we do not need to list. Lemma 1. Let p be a prime with p ≡ 1 mod 6. en for any six-order character ψ mod p, we have the identity Proof. For this, refer the study of Chen [11].
□ Lemma 2. Let p be a prime with p ≡ 7 mod 12, χ 2 denote Legendre's symbol modulo p, and λ denote any three-order Dirichlet character modulo p. en for any integer m with (m, p) � 1, we have the identities 2

Journal of Mathematics
Proof. It is clear that for any integer r with (r, p) � 1, from the properties of the three-order character modulo p, we have It is clear that χ 2 � χ 2 is a real character modulo p; from (15) and the properties of the classical Gauss sums, we have Note that p ≡ 3 mod 4, From (16) and (17) and Lemma 1, we have Proof. Note that p ≡ 1 modulo 4, λ 2 � λ, and τ(χ 2 λ)τ(χ 2 λ) � p; from (16) and the methods of proving Lemma 2, we also have (20) It is clear that Lemma 3 follows from (20)-(22).

Proofs of the Theorems
Now we shall complete the proofs of our all results. First we prove eorem 1. Let p be an odd prime with p ≡ 7 mod 12, and then note that τ 2 (χ 2 ) � − p; from Lemma 2 and the properties of the character sums modulo p, we have From (23) and Lemmas 1 and 2, we have If n ≥ 3, then 2n ≥ 6, from Lemma 2, we have From (23)-(26) and the definition of G n (p), we may immediately deduce the three-order linear recursion formula with the three initial values G 0 (p) � p − 1, G 1 (p) � − 3(p − 1)p, and is proves eorem 1. Now we prove eorem 2. If p be an odd prime with p ≡ 1 mod 12, then note that χ 2 (− 1) � 1 and τ(χ 2 ) � � � p √ ; from Lemma 3, we have It is clear that from Lemmas 1 and 3, we have From (30)-(32) and the definition of G n (p), we have the three-order linear recursion formula Journal of Mathematics where the three initial values G 0 (p) � p − 1, G 1 (p) � 3 (p − 1)p, and G 2 (p) � (p − 1)p(11p + 4d 2 ).

Conclusion
e 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 p with p ≡ 1 mod 3, we have the identities ese 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
e author declares that there are no conflicts of interest regarding the publication of this paper.

Authors' Contributions
e author contributed to the work and read and approved the final manuscript.