A Note on the Classical Gauss Sums

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 mod p ( a prime with p ≡ 1 mod 12) to give an exact evaluation formula for it.


Introduction
As usual, let q ≥ 3 be an integer, χ be any Dirichlet character mod q.Then the classical Gauss sum τ(χ) is defined as follows: where e(y) = e 2πiy .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 [1] provided the following result: 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 indicates the the Legendre symbol mod p, and α denotes 1 2 ∑ p−1 a=1 a+a p for convenience.H. Bai and J. Y. Hu [2] 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 p ≡ 1 mod 8, ψ be any eighth-order character mod p, and Then we have the second-order linear recursive formula where F 0 (p) = 2, F 2 (p) = 2α √ p , and α is the same as in (1).W. P. Zhang and J. Y. Hu [3] (the different form can also be found in B. C. Berndt and R. J.Evans [4] ) proved that for any prime p with p ≡ 1 mod 3 and any third-order character ψ mod p, one has 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.
Chen Li [5] used the identity (2) to prove the following conclusions.Let Then for any prime p with p ≡ 1 mod 12 and any third-order character ψ mod p, one has the second-order linear recursive formula where the initial values U 0 (p, ψ) = 2 and U 1 (p, ψ) = d 2 −2p p , d is the same as in (1).Therefore, from (3) one can deduce the general term If p be a prime with p ≡ 7 mod 12, then one has where the initial values p .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 [6][7][8][9][10][11][12][13][14][15] .Here we will not list them one by one.
In this paper, as a note of [2,5], we will study a similar problem for prime p is a prime with p ≡ 1 mod 12 and any twelfth-order character χ mod p.More specifically, let χ be any twelfth-order character mod p 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 p ≡ 1 mod 12, χ be any twelfth-order character mod p. Then for any positive integer k, we have the second-order linear recursive formula where the initial values W , d is uniquely determined by 4p = d 2 + 27b 2 and d ≡ 1 mod 3.
From this recursive formula we may immediately deduce the general term .

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 mod p, all of which can be found reference [16], so we will not repeat them here.First we have the following: Lemma 1.Let p be a prime with p ≡ 1 mod 6, χ be any sixth-order character mod p. Then about the classical Gauss sums τ(χ), the following identity holds, where i 2 = −1, d is uniquely determined by 4p = d 2 + 27b 2 and d ≡ 1 mod 3.
Proof.In fact, this is Lemma 3 of [5], so its proof is omitted.
Lemma 2. Let p be a prime with p ≡ 1 mod 12. Then for any twelfth-order character χ mod p, we obtain the identity τ 6 (χ) τ 6 (χ 5 ) where d is the same as in (2).
Proof.Let ψ be any third-order character mod p, λ be any fourth-order character mod p. Then for any twelfth-order character χ mod p, we must have χ = ψλ, ψλ, ψλ or ψλ.Without loss of generality we suppose that χ = ψλ, then for any integer b with (b, p) = 1, note that we have the identity (see Theorem 7.5.4 in [17]) where χ 2 = * p denotes the Legendre's symbol mod p.

Proof of the Theorem
In this section, we complete the proof of our theorem.In fact for any prime p with p ≡ 1 mod 12 and any twelfth-order character χ mod p, from the definition of W k (p, χ) and Lemma 2 we have The formula (11) implies that = β.This completes the proof of our theorem.

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 [2,5].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 χ mod p, then p just has φ(12) = 4 different twelfth-order characters χ, χ, χ 5 and χ 5 mod p, they all appear in (5), and conjugated configuration.
Then we think there must be a positive integer r so that H k (p, χ) has a second-order linear recurrence formula as in our theorem.In fact, for a certain positive integer r, we only need to know H 1 (p, χ).
This still is an open problem.We will continue to study.