On the Hybrid Power Mean of Two-Term Exponential Sums and Cubic Gauss Sums

Shaofan Cao and Tingting Wang College of Science, Northwest A&F University, Yangling, Shaanxi, China Correspondence should be addressed to Tingting Wang; ttwang@nwsuaf.edu.cn Received 14 December 2020; Revised 31 March 2021; Accepted 10 May 2021; Published 30 May 2021 Academic Editor: Stanislaw Migorski Copyright © 2021 Shaofan Cao and Tingting Wang. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited. In this paper, an interesting third-order linear recurrence formula is presented by using elementary and analytic methods. *is formula is concerned with the calculating problem of the hybrid powermean of a certain two-term exponential sums and the cubic Gauss sums. As an application of this result, some exact computational formulas for one kind hybrid power mean of trigonometric sums are obtained.


Introduction
As usual, let p be an odd prime. For any integer m, we define the cubic Gauss sums A(m, p) � A(m) as follows: where e(y) � e 2πiy and i 2 � − 1.
In recent years, several scholars have studied the hybrid power mean problems of various trigonometric sums and proved many interesting results. For example, Chen and Hu [1] studied the computational problem of where c denotes the multiplicative inverse of c mod p. at is, c · c ≡ 1 mod p. For p ≡ 1 mod 3, they obtained a thirdorder linear recurrence formula for S k (p). Li and Hu [2] studied another hybrid power mean and gave an exact computational formula for (3). Some other related papers can also be found in references [3][4][5][6][7][8][9][10][11]. We would not have to repeat them here.
Very recently, Chen and Chen [12] studied the recursive properties of the hybrid power mean and obtained a third-order linear recurrence formula for it with c � 1 and p ≡ 1 mod 3. at is, they proved the following result: Let p be an odd prime with p ≡ 1 mod 3. If 3 is a cubic residue mod p, then for any integer k ≥ 3, one has the thirdorder linear recurrence formula: where the first three terms are H 0 (1, p) � 2p 2 − pd, Note that Zhang and Zhang [13] proved the identity Perhaps this is the best result for which there is no conditional requirement on the prime p. e interesting results of the above work motivate us to ask such a problem of whether there exists a similar recursive formula for the hybrid power mean where p is an odd prime with p ≡ 1 mod 3.
Obviously, the problem in (7) is much harder than the problem in [12] because we are dealing with the fourth power mean of the two-term exponential sums in (7). Our main contribution is to obtain an identity for the fourth power mean of the two-term exponential sums weighted by third-order character modulo p, i.e., the following Lemma 3. en, we use this lemma to derive several interesting recursion formulas for U k (p). In this way, the continuities and the value distribution properties of this kind of trigonometric sums can be described from different views. Of course, the reason why we focus on the calculation of (7) is that the problem is closely related to the number of the solutions of some congruence equation. ese contents play a very important role in study of some famous analytic number theory problems, such as Waring problem and Goldbach conjecture.
rough the study, it is found that the problem we studied is closely related to integer 3. If 3 is a cubic residue modulo p, then there exists a beautiful third-order linear recurrence formula for U k (p), and the first three terms U 0 (p), U 1 (p), and U 2 (p) are integers. If 3 is not a cubic residue mod p, then we can get the exact value of U 2 (p). For any other positive integer k, we can only give a more complex mathematical representation for U k (p). at is, we have the following three results: Theorem 1. Let p be a prime with p ≡ 1 mod 3. If 3 is a cubic residue modulo p, then for any integer k ≥ 3, we have the third-order linear recurrence formula where the first three terms are and d is uniquely determined by 4p � d 2 + 27b 2 and d ≡ 1 mod 3.

Theorem 2.
Let p be a prime with p ≡ 1 mod 3. If 3 is a cubic residue modulo p, then for any integer k ≥ 3, we also have the third-order linear recurrence formula where the first three terms are Theorem 3. Let p be an odd prime with p ≡ 1 mod 3. If 3 is not a cubic residue modulo p, then we have the identity From our theorems, we may immediately deduce the following three corollaries: Corollary 1. Let p be an odd prime with p ≡ 1 mod 3, then we have Corollary 2. Let p be a prime with p ≡ 1 mod 3. If 3 is a cubic residue modulo p, then we have 2 Journal of Mathematics Corollary 3. Let p be a prime with p ≡ 1 mod 3. If 3 is a cubic residue modulo p, then we have Some notes: first in eorem 2, if (3, p − 1) � 1, then the question we are discussing is trivial. Because in this case, we have Second, if p ≡ 1 mod 3 and 3 is not a cubic residue modulo p, then we can only get the exact value of U 2 (p).
Third, the advantage of our work is that we completely solve the calculation problem of U k (p) with p ≡ 1 mod 3.
Fourth, the mean value estimation of the exponential sums is closely related to the upper and lower bounds of the individual exponential sums. So, by studying the mean value of the positive exponential sums, we can obtain a better upper bound estimation of the exponential sums. If we want to get its lower bound estimation of the exponential sums, we should study the negative power of the exponential sums. Our Theorems 1 and 2 address both types of problems.
Finally, for any fixed positive integer h ≥ 5, whether there is a third-order linear recurrence formula for the hybrid power mean is an open problem, which is the limitation of our work. e other drawback, of course, is that we cannot compute all U k (p) when 3 is not a cubic residue modulo p. In fact, our ultimate goal is to obtain a precise calculation formula for W k (h, p) for all positive integers h ≥ 5. In the future, we will continue to improve the research in this aspect. It also requires us to continue to study.

Several Lemmas
To complete the proofs of our theorems, several simple lemmas are necessary. Hereafter, we will use many properties of the classical Gauss sums and the third-order character modulo p, all of which can be found in books concerning about Elementary Number eory or Analytic Number eory, such as references [14][15][16], so the related contents will not be repeated here. First we have the following:

Lemma 1. If p is a prime with p ≡ 1 mod 3, then for any third-order character ψ mod p, we have the identity
Proof. First applying trigonometric identity and noting that ψ 3 � χ 0 , the principal character modulo p, we have

(18)
Note that ψ 2 � ψ and τ(ψ)τ(ψ) � p, and from the properties of Gauss sums and the characteristic function of the third-order character modulo p we have Since ψ is a third-order character modulo p, for any integer c with (c, p) � 1, from the properties of the classical Gauss sums, we have From (22) and the properties of Gauss sums, then we can get Combining (18), (20), (21), and (23), we have the identity (24) is proves Lemma 1. □ Lemma 2. If p is a prime with p ≡ 1 mod 3 and ψ is any third-order character modulo p, then we have the identity Proof. See [3] or [11].
□ Lemma 3. If p is a prime with p ≡ 1 mod 3, then for any third-order character ψ mod p, we have the identity Proof. Note that the two-term exponential sums satisfies So, from the properties of Gauss sums and Lemma 1, we have (28)

(29)
It is clear that A(m) is a real number, so from the properties of Gauss sums and (29), we have Note that the congruence a + b ≡ c + 1 mod p implies the congruence Combining (28)