On Classical Gauss Sums and Some of Their Properties

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.


Introduction
Let q ≥ 3 be an integer.For any Dirichlet character χ mod q, according to the definition of classical Gauss sums τ(χ), we can write where e(y) = e 2πiy .
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 [1]: Let p be an odd prime with p ≡ 1 mod 4, λ be any fourth-order character mod p. Then one has the identity where * p = χ 2 denotes the the Legendre's symbol mod p (please see Reference [1,2] for its definition and related properties), and If p is a prime with p ≡ 1 mod 3, ψ is any third-order character mod p, then Zhang and Hu [3] had already obtained an analogous result (see Lemma 1).However, perhaps the most beautiful and important property of Gauss sums τ(χ) is that |τ(χ)| = √ q, for any primitive character χ mod q.
where p is an odd prime, k is a non-negative integer, χ is any non-principal character mod p.
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 U k (p, χ) has some good properties.In fact, for some special character χ mod p, the second-order linear recurrence formula for U k (p, χ) for all integers k ≥ 0 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 U k (p, χ), and to calculate two recursive formulae, which are listed hereafter: Theorem 1.Let p be a prime with p ≡ 1 mod 12, ψ be any third-order character mod p.Then, for any positive integer k, we can deduce the following second-order linear recursive formulae where the initial values U So we can deduce the general term Theorem 2. Let p be a prime with p ≡ 7 mod 12, ψ be any third-order character mod p.Then, for any positive integer k, we will obtain the second-order linear recursive formulae where the initial values U p and i 2 = −1.
Similarly, we can also deduce the general term

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 mod p, all of which can be found in books concerning elementary and analytic number theory, such as in References [2,10], so we will not duplicate the related contents.Lemma 1.If p is any prime with p ≡ 1 mod 3, ψ is any third-order character mod p, then, we have the equation where τ (ψ) denotes the classical Gauss sums, d is uniquely determined by 4p = d 2 + 27b 2 and d ≡ 1 mod 3.
Lemma 2. Let p be a prime with p ≡ 1 mod 3, ψ be any third-order character mod p, χ 2 = * p denotes the Legendre's symbol mod p.The following identity holds Proof.Firstly, using the properties of Gauss sums, we get ( On the other side, we get the sums Combining Equations ( 2) and (3), we obtain Now, Lemma 2 has been proved.
Lemma 5. Let p be a prime with p ≡ 1 mod 12, ψ be any three-order character mod p.Then, we obtain the sum term Proof.From Lemma 3 and the method of proving Lemma 4 we can easily deduce Lemma 5.

Proofs of the Theorems
In this section, we prove our two theorems.For Theorem 1, since p ≡ 1 mod 12, ψ is a third-order character mod p, 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 U 0 (p) = 2 and U 1 (p) = d 2 −2p p .Note that the two roots of the equation λ 2 − d 2 −2p p λ + 1 = 0 are .