On the sixth power mean of one kind two-term exponential sums weighted by Legendre's symbol modulo $p$

where as usual, e(y) = e2πiy and i2 = −1. Since this kind sums play an very important role in the study of analytic number theory, so many number theorists and scholars had studied the various properties of C(m, k; q), and obtained a series of meaningful research results, we do not want to enumerate here, interested readers can refer to references [2–7,9–11,13–15]. Note that |C(m, k; q)| is a multiplicative function of q, so people often only consider case that q = p or pr, where p is an odd prime, and r ≥ 2 is a positive integer. For example, H. Zhang and W. P. Zhang [13] proved that for any odd prime p, one has


Introduction
Let q ≥ 3 be a fixed integer. For any integer k ≥ 2 and m with (m, q) = 1, we define the two-term exponential sums C(m, k; q) as follows: where as usual, e(y) = e 2πiy and i 2 = −1.
Since this kind sums play an very important role in the study of analytic number theory, so many number theorists and scholars had studied the various properties of C(m, k; q), and obtained a series of meaningful research results, we do not want to enumerate here, interested readers can refer to references [2][3][4][5][6][7][9][10][11][13][14][15]. Note that |C(m, k; q)| is a multiplicative function of q, so people often only consider case that q = p or p r , where p is an odd prime, and r ≥ 2 is a positive integer.
For example, H. Zhang and W. P. Zhang [13] proved that for any odd prime p, one has where n represents any integer with (n, p) = 1.
L. Chen and X. Wang [3] studied the calculating problem of the fourth power mean of G(m, 4; p), and proved the following conclusion: if p = 12k + 7; 2p p 2 − 4p − 2α 2 if p = 24k + 5; 2p p 2 − 6p − 2α 2 if p = 24k + 13; 2p a + a p , and * p denotes the Legendre's symbol modulo p, and a · a ≡ 1 mod p. Z. Y. Chen and W. P. Zhang [6] proved that for any prime p with p ≡ 5 mod 8, one has the identity where α = α(p) is the same as defined in the above. Very recently, J. Zhang and W. P. Zhang [14] studied the fourth power mean of the two-term exponential sums weighted by Legendre's symbol modulo an odd prime p, and proved that for any odd prime p, one has the identity The main purpose of this paper as a generalization of (3.1), and study the calculating problem of the 2h-th power mean of the two-term exponential sums where p is an odd prime, and h ≥ 3 is an integer. It is clear that J. Zhang and W. P. Zhang [14] proved an identity for G(2, p). But for h ≥ 3, it seems that none had studied it before, at least we have not seen such a result at present. We think this content is meaningful for further research. Because it can solve the problem of calculating the 2h-th power mean In other words, we shall deal with the 2h-th power mean problem involving the sums of quadratic residues modulo p. This will provide some new ideas and methods for us to study the power mean problem on special sets. Of course, the problem we are studying in here is much more difficult than that in [14], because we are going to do the sixth power mean, some congruence equations involved more variables, this can lead to the computational difficulties.

Several lemmas
In this section, we will give several necessary lemmas. Of course, the proofs of some lemmas need the knowledge of elementary and analytic number theory. In particular, the properties of the quadratic residues and the Legendre's symbol modulo p. All these can be found in references [1,8,12], and we do not repeat them. First we have Lemma 1. Let p > 3 be an odd prime. Then we have the identity , from the properties of the complete residue system modulo p we have This proves Lemma 1. Lemma 2. Let p be an odd prime, then we have the identity Proof. From the properties of the complete residue system and quadratic residue modulo p we have For any integer n, note that the identity Combining (2.1), (2.2) and the properties of the quadratic residue modulo p we have This proves Lemma 2. Lemma 3. Let p be an odd prime, then we have the identity Proof. From the properties of the complete residue system modulo p we have This proves Lemma 3.
Proof. From the properties of the complete residue system modulo p we have From (2.2) and the properties of the quadratic residue modulo p we have It is easy to prove that This proves Lemma 4.