On the hybrid power mean of two kind different trigonometric sums

Abstract The main purpose of this paper is using the analytic method, the properties of trigonometric sums and Gauss sums to study the computational problem of one kind hybrid power mean involving two different trigonometric sums, and give an interesting computational formula for it.


Introduction
Let p be an odd prime. The quartic Gauss sums C(m, p) = C(m) is de ned as where as usual, e(y) = e πiy . Recently, several scholars studied the properties of C(m, p), and obtained some interesting results. For example, Shen Shimeng and Zhang Wenpeng [1] obtained a fourth-order linear recurrence formula for C(m, p).
Li Xiaoxue and Hu Jiayuan [2] studied the computational problem of the hybrid power mean and proved an exact computational formula for (1), where c denotes the multiplicative inverse of c mod p.
That is, c · c ≡ mod p. In the same paper [2], the authors also suggested us to calculate the exact value of the Gauss sums where p ≡ mod be a prime, k be any positive integer, ψ denotes a fourth-order character mod p, and τ(ψ) denotes the classical Gauss sums. That is, Chen Zhuoyu and Zhang Wenpeng [3] used the analytic method and the properties of the classical Gauss sums to obtain an interesting recurrence formula for G(k, p), which completely solved the computational problem of G(k, p). Some works related to the power mean of the trigonometric sums can also be found in references [4]- [8]. They will not be repeated here. Inspired by references [1], we will consider the following hybrid power mean We naturally ask: does there exist a precise computational formula for (2)? The main purpose of this paper is to answer this question. For convenience, we assume that p is a prime with p ≡ mod , * p = χ denotes the Legendre symbol mod p, and This α closely related to prime p. In fact, we have the Square Sum Theorem: where r is any quadratic non-residue mod p (see Theorem 4-11 in [9]). In this paper, we will use the properties of Gauss sums and Legendre symbol to study the computational problem of (2), and give an interesting computational formula for it. That is, we will prove the following two conclusions. Theorem 1. If p is a prime with p ≡ mod , then we have the identity

Theorem 2.
If p is a prime with p ≡ mod , then we have where ψ is any fourth-order character mod p, and |G( , Note that the estimations |G( , p)| ≤ √ p and |α| ≤ √ p, from our theorems we may immediately deduce the following two corollaries: Corollary 1. Let p be an odd prime with p ≡ mod , then we have the asymptotic formula where k is any positive integer. Only the calculation is more complex, if k is large enough. So we have not given a general conclusion here.

Several Lemmas
To complete the proofs of our theorems we need four simple lemmas. Here we will use many properties of the classical Gauss sums and Legendre's symbol mod p, all of them can be found in many elementary number theory books, such as reference [11], so the related contents will not be repeated here. First we have the following: Lemma 1. If p is a prime with p ≡ mod , then for any fourth-order character ψ mod p, we have the identity Similarly, we can also deduce that On the other hand, from the properties of the fourth-order character ψ mod p we have where we have used identity ψ (a) = for any integer a with (a, p) = .