New Identities Dealing with Gauss Sums

: In this article, we used the elementary methods and the properties of the classical Gauss sums to study the problem of calculating some Gauss sums. In particular, we obtain some interesting calculating formulas for the Gauss sums corresponding to the eight-order and twelve-order characters modulo p , where p be an odd prime with p = 8 k + 1 or p = 12 k + 1.


Introduction
For any integer q > 1 and any Dirichlet character χ modulo q, the famous Gauss sums G(m, χ; q) is defined as follows: where m is any integer and e(y) = e 2πiy . If χ is any primitive character modulo q or m co-prime to q (that is, (m, q) = 1), then we have the identity G(m, χ; q) = χ(m)G(1, χ; q) ≡ χ(m)τ(χ).
If χ is a primitive character modulo q, then for any integer m, we also have the following two important identities: χ(b) e mb q and |τ(χ)| = √ q.
As it known to all, the research on the properties of Gauss sums occupies very important position in analytic number theory, many number theory problems are closely related to it. Because of this, many scholars have studied its various properties, and obtained a number of interesting results, some of them and related works can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In addition, Gauss sums are closely related to prime numbers. For example, if p is an odd prime with p ≡ 1 mod 3, then there are two integers d and b such that the identity (see [7]) holds where d is uniquely determined by d ≡ 1 mod 3. Zhang, W.P. and Hu, J.Y. [1] or Berndt, B.C. and Evans, R.J. [8] studied the properties of Gauss sums of the third-order character modulo p, and proved the following result: Let p be a prime with p ≡ 1 mod 3. Then for any third-order character χ 3 mod p, one has the identity where d is the same as defined in (1). Chen, Z.Y. and Zhang, W.P. [3] 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. Then for any fourth-order character χ 4 mod p, one has the identity where * p = χ 2 denotes the Legendre's symbol mod p and aa ≡ 1 mod p. And of course, α in (3) can also be represented by quadratic Gauss sum. Chen, L. [4] studied the properties of the Gauss sums of the sixth-order character modulo p, and deduced an interesting identity (see Lemma 1 below).
Looking closely at the characteristics of these results, it is not difficult to see that the number of all such characters satisfy φ(3) = φ(4) = φ(6) = 2, where φ(n) denotes the Euler function. So a natural thing to think about is, what if the number of the characters > 2? For example, twelfth-order character modulo p with φ (12) In this paper, we shall use the properties of the classical Gauss sums, the elementary and analytic methods to study this problem, and obtain two interesting identities for them. That is, we shall prove the following two results: Theorem 1. Let p be an odd prime with p ≡ 1 mod 8. Then for any eighth-order character χ 8 modulo p, we have the identity where α = α(p) is the same as defined in (3).

Theorem 2.
Let p be an odd prime with p ≡ 1 mod 12. Then for any third-order character λ and fourth-order character χ 4 modulo p, we have the identity where d is the same as defined in (1).
From these two theorems we may immediately deduce the following identities: If p be an odd prime with p ≡ 1 mod 8, then for any eighth-order characters χ 8 modulo p, we have the identities Corollary 2. If p is an odd prime with p ≡ 1 mod 12, then for any third-order character λ and fourth-order characters χ 4 modulo p, we have the identities Some notes: Since λχ 4 is a twelfth-order character modulo p in Theorem 2, so our Theorem 1 and Theorem 2 extend the results in references [1,3,4].
The constant α = α(p) in Theorem 1 has a special meaning. In fact, if p ≡ 1 mod 4, then we have the identity (see Theorem 4-11 in [18]) where r is any quadratic non-residue modulo p. That is, r p = −1.

Several Lemmas
In this section, we give several simple lemmas. Of course, the proofs of these lemmas need some knowledge of elementary and analytic number theory. They can be found in many number theory books, such as [18,19], here we do not need to list. First we have the following: Lemma 1. Let p be a prime with p ≡ 1 mod 6. Then for any sixth-order character ψ mod p, one has the identity Proof. This result is Lemma 3 in Chen, L. [4], so we omit the proof process.

Lemma 2.
Let p be a prime with p ≡ 1 mod 12. Then for any third-order character λ and fourth-order character χ 4 modulo p, we have the identity where * p = χ 2 denotes the Legendre's symbol mod p.
Proof. Let ψ = λχ 4 be any twelfth-order character modulo p, where λ is a third-order character and χ 4 is a fourth-order character modulo p respectively. Then note that ψ 2 = λ 2 χ 2 4 = λχ 2 , from the properties of Gauss sums we have On the other hand, note that for any integer b with (b, p) = 1, we have the identity so note that λ(−1) = 1 we also have the identity Combining (4) and (5) we have the identity This proves Lemma 2.
Lemma 3. Let p be a prime with p ≡ 1 mod 8. Then for any eighth-order character χ 8 modulo p, we have the identity Proof. Let χ 8 be an eighth-order character modulo p, then from the properties of Gauss sums we have On the other hand, we also have the identity Combining (6) and (7) we have the identity This proves Lemma 3.