On two-term exponential sums and their mean values

: The mean value problems of exponential sums play a very important role in the research of analytic number theory, and many famous number theory problems are closely related to them. The main purpose of this paper is using some elementary methods and the number of the solutions of the congruence equations to study the calculating problem of some fourth power means of two-term exponential sums, and give exact calculating formulae and asymptotic formula for them.


Introduction
As usual, let p be an odd prime. For any integers h > r ≥ 1 and integers m and n, the two-term exponential sum S (m, n, h, r; p) is defined as S (m, n, h, r; p) = p−1 a=0 e ma h + na r p , where e(y) = e 2πiy and i is the imaginary unit. That is, i 2 = −1.
These sums play very important role in the study of analytic number theory, so many number theorists have studied the various properties of S (m, n, h, r; p) and obtained some meaningful research results. Here we give a few examples. W. P. Zhang and D. Han [1] used analytic methods to study the sixth power mean of S (1, m, 3, 1; p), and proved the following result: where p denotes an odd prime with 3 ∤ (p − 1).
Recently, W. P. Zhang and Y. Y. Meng [2] also studied the sixth power mean of S (m, n, 3, 1; p), and obtained some new identities. In fact, they proved that for any odd prime p and integer n with (n, p) = 1, one has p−1 m=1 p−1 a=0 e ma 3 + na p ( 1.2) here, 4p = d 2 + 27 · b 2 , and d is uniquely determined by d ≡ 1 mod 3.
In addition, some related papers can also be found in [5][6][7][8][9][10][11]. From the formulae (1.1)-(1.3), it is not difficult to see that the content of all these papers only involves r = 1 in S (m, n, h, r; p). For the case r > 1 in S (m, n, h, r; p), we have not found a corresponding conclusion, at least not so far. Therefore, when h > r = 2, the research is difficult, and it is difficult to obtain some ideal results.
In this paper, we use elementary methods and results on the number of solutions to study the calculating problem of the 2k-th power mean S 2k (p) = Some notes: In Theorem 1, we only discussed a prime p in the case p ≡ 3 mod 4. If p ≡ 1 mod 4, then the situation is much more complicated. We cannot calculate the exact value of S 4 (p), and we cannot even get a valid asymptotic formula for S 4 (p). The reason is that we do not know the exact value of and do not know a valid asymptotic formula for it. However, if p ≡ 1 mod 4, we give some ideas and methods of studying S 4 (p) (see Lemma 3 and Lemma 4 below). We even have the following: Conjecture. If prime p ≡ 1 mod 4, then there is an asymptotic formula To prove this conjecture, we can convert (1.4) to the estimate for character sums If one can give a non-trivial upper bound estimate for D(p), such as |D(p)| ≪ p, then we can prove that the conjecture is correct.
For any odd prime p with p ≡ 3 mod 4 and integer k ≥ 3, whether there exists an exact calculating formula for is an open problem. Interested readers can continue this research.

Several lemmas
To complete the proofs of our main results, we need some simple lemmas. Of course, the proofs of all these lemmas need some knowledge of elementary number theory and analytic number theory, and all these can be found in [12,13], so we do not have to repeat them here. First, we have the following: Lemma 1. For any odd prime p, we have the identities Proof. It is clear that Similarly, we also have Proof. First, we prove the second formula in Lemma 2. Let b denote the inverse of b mod p, i.e., bb ≡ 1(mod p). From the properties of a complete residue system modulo p, we have Since p ≡ 3 mod 4, from the properties of the Legendre symbol modulo p, we have −1 p = −1 and Similarly, we also have It is clear that we have the identity Similarly, we can also deduce the first formula in Lemma 2, which is Proof. The proof of the lemma is mainly divided into three parts: (i) Summing Legendre symbols containing one variable, we have Note the identities We have    Proof. Let The proof of the lemma is divided into the following four parts: (i) There is no Legendre symbol after S . From a complete residue system modulo p, we have (ii) Legendre symbol contains one variable after S . Then, we can get (2.14) (iii) Legendre symbol contains one variable after S . Then, we can get This proves Lemma 4. □

Proofs of the Theorems
Now, we apply Lemmas 1-4 in Section 2 to complete the proofs of our theorems. For any positive integer q > 1 and integer n, note the trigonometric identities

Conclusions
The main result of this paper is to give an exact calculating formula for the fourth power mean of one special two-term exponential sum. That is, with the case p ≡ 3 mod 4. If p ≡ 1 mod 4, then we cannot get any non-trivial results for S 4 (p) yet. We also point out the key to the problem, which is we do not know the exact value of and do not know a valid asymptotic formula for it. However, we get some periodic results. At the same time, some problems to be further studied are also proposed. In any case, our work provides a new method for the study of relevant problems. We have a reason to believe that this work will play a positive role in promoting the study of relevant problems.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.