On the Hybrid Fourth Power Mean Involving Legendre’s Symbol and One Kind Two-Term Exponential Sums

where, as usual, e(y) � e2πiy and i denotes the imaginary unit, that is i2 � − 1. Since this kind of sums play a very important role in the study of analytic number theory, so many number theorists and scholars had studied the various properties of G(m, k; q) and obtained a series of meaningful research results, we do not want to enumerate here, and interested readers can refer to [1–16]. For example, Zhang and Zhang [1] proved that for any odd prime p, one has


Introduction
Let q ≥ 3 be a fixed integer. For any integer k ≥ 2 and integer m with (m, q) � 1, we define the two-term exponential sums G(m, k; q) as follows: where, as usual, e(y) � e 2πiy and i denotes the imaginary unit, that is i 2 � − 1.
Since this kind of sums play a very important role in the study of analytic number theory, so many number theorists and scholars had studied the various properties of G(m, k; q) and obtained a series of meaningful research results, we do not want to enumerate here, and interested readers can refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. For example, Zhang and Zhang [1] proved that for any odd prime p, one has where n represents any integer with (n, p) � 1. Shen and Zhang [2] obtained an interesting recurrence formula for where p is an odd prime with p ≡ 1 mod 4. Chen and Zhang [3] proved that for any prime p with p ≡ 5 mod 8, one has the identity (a + a/p), ( * /p) denotes Legendre's symbol modulo p, and a · a ≡ 1 mod p.
Zhang and Han [4] used the elementary method to obtain the identity where p denotes an odd prime with 3∤(p − 1). Chen and Wang [5] studied the calculating problem of the fourth power mean of G(m, 4; p) and proved the following conclusion.
Let p > 3 be an odd prime, then one has the identity (a + a/p) and ( * /p) denotes Legendre's symbol modulo p.
Zhang and Zhang [6] proved that for any prime p, one has the identity Liu and Zhang [7] proved that for any prime p with 3 ∤(p − 1), one has the identity where χmodp denotes the summation over all Dirichlet characters, modulo p. Inspired by the works in [5,6], in this paper, we consider the following calculating problem of the 2h-th power mean of the two-term exponential sums: where p is an odd prime and h ≥ 2 is an integer.
About this problem, it seems that none had studied it before; at least we have not seen such a result at present. In this paper, we will use the properties of the solutions of the congruence equations and the quadratic residue to study this problem and give an interesting calculating formula for (9) with h � 2. at is, we will prove the following result.

Theorem 1. Let p > 3 be an odd prime, then we have the identity
(a + a/p) and ( * /p) denotes Legendre's symbol modulo p.
Note that the estimate |α| ≤ � � p √ (see [17] for general results), and from this theorem and [5], we may immediately deduce the following several corollaries. Corollary 1. Let p > 3 be an odd prime with p ≡ 1 mod 4, then we have the asymptotic formula Some notes: the constant α � α(p) in our theorem has a special meaning. In fact, for any prime p with p ≡ 1mod4, one has the identity (see [18]) where r is any quadratic nonresidue modulo p. at is, For any prime p � 4k + 1 and integer h ≥ 3, we naturally ask whether there is an exact calculating formula for (9)? is is an open problem. We believe this to be true. We even have the following.

Conjecture 1.
Let p be an odd prime with p ≡ 1mod4. en, for any integer h ≥ 2, there are two integers C � C(h, p) and D � D(h, p) depending only on h and p, such that the identity

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 are going to be used. All these can be found in [15,[18][19][20], and we do not repeat them. First, we have the following lemma.
Proof. See Lemma 2 in Chen and Zhang [3]. □ Lemma 2. Let p be an odd prime with p ≡ 1mod4, then we have the identity where α � α(p) is defined as in Lemma 1.