The Recursive Properties of the Error Term of the Fourth Power Mean of the Generalized Cubic Gauss Sums

q 􏼠 􏼡, (2) where k is any positive integer and m is an integer with (m, q) � 1. It is clear that this sum is a generalization of the classical Gauss sums G(m, χ; q). In fact, G(m, 1, χ; q) � G(m, χ; q). Of course, the value of G(m, k, χ; q) is irregular as χ varies. However, some scholars have found that G(m, k, χ; q) has good value distribution properties in some problems of weightedmean value, even if we can get their exact calculation formulae for some 2kth power mean. In addition, there are some good upper bound estimates for |G(m, k, χ; q)|. For example, for any integer n with (n, q) � 1, from the general result of Cochrane and Zheng [1], we can deduce


Introduction
For any integer q ≥ 2 and any Dirichlet character χ mod q, the definition of the classical Gauss sums G(m, χ; q) is where m is an integer and e(y) � e 2πiy . is sum and its properties are of great significance to the analytic number theory, and many number theory problems are closely related to them. erefore, it is necessary to study the various properties of G(m, χ; q) and related sums. In this paper, we consider the generalized k-th Gauss sums G(m, k, χ; q), which is defined as follows: where k is any positive integer and m is an integer with (m, q) � 1. It is clear that this sum is a generalization of the classical Gauss sums G(m, χ; q). In fact, G(m, 1, χ; q) � G(m, χ; q). Of course, the value of G(m, k, χ; q) is irregular as χ varies. However, some scholars have found that G(m, k, χ; q) has good value distribution properties in some problems of weighted mean value, even if we can get their exact calculation formulae for some 2kth power mean. In addition, there are some good upper bound estimates for |G(m, k, χ; q)|. For example, for any integer n with (n, q) � 1, from the general result of Cochrane and Zheng [1], we can deduce where ω(q) denotes the number of distinct prime divisors of q. e case that q is a prime is due to Weil [2]. For k � 2, by the results of Zhang [3], let n be any integer with (n, p) � 1, and there are the following two identities: where ( * /p) denotes the Legendre symbol modulo p. Zhang and Liu [4] have studied the sum χmodp |G(n, 3, χ; q)| 4 (5) and obtained the following calculation formula: where p is a prime with 3|p − 1 and U � p a�1 e(a 3 /p) is a real constant.
However, the value of U was not given in [4], and the form of U was not concise. Now, for any integer In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the nth power mean of E(m, p) and give two recurrence formulae for it. at is, we shall prove the following main results.

Theorem 2.
Let p be an odd prime with 3|p − 1. en, for any positive integer n ≥ 3, we have the recurrence formula where the first three terms of (11) Theorem 3. Let p be an odd prime with 3|p − 1. en, for any positive integer n ≥ 1, we have the third-order linear recurrence formula where the initial values of Taking n � 4 in eorem 3, we may immediately deduce the following corollary. Corollary 1. Let p be an odd prime with 3|p − 1; then, we have the identity

Several Lemmas
In this section, we give three lemmas which are necessary in the proofs of our theorems. In the process of proving our lemmas, we need some knowledge of the analytic number theory; all of which can be found in [6][7][8], so it is not necessary to repeat them here.

Lemma 1.
Let p be an odd prime with p ≡ 1mod3. en, for any third-order character λ mod p, we have where d is uniquely determined by 4p � d 2 + 27b 2 and d ≡ 1mod3.
Proof. is result can be found in [9] or [10].
Proof. For any integer 1 ≤ a ≤ p − 1, it is easy to show that From the properties of the cubic character modulo p, we have Journal of Mathematics So, we have the identity erefore,
□ Lemma 3. Let p be an odd prime with 3|p − 1. en, for any cubic character λ modulo p, we have the identity where and E(m, p) is defined as the same as in Lemma 2.

Proofs of the Theorems
Now, we shall complete the proofs of our main results. Firstly, we prove eorem 1. Let p be an odd prime with 3|p − 1, χ be any Dirichlet character modulo p, and λ be a cubic character modulo p. en, from the properties of the classical Gauss sums and (17), we have where E(m, p) is the same as in Lemma 2.
For any integer n, we have the trigonometric identity

Journal of Mathematics
From (30), we have the identity where U(p, d) � 64 dp + d 5