A Hybrid Power Mean Involving the Dedekind Sums and Cubic Gauss Sums

where as usual, e(x) � e2πix and i2 � − 1. *is sum plays a very important role in the study of the elementary number theory and analytic number theory, so there are many people who had studied the arithmetical properties of A(m, q) and related contents (see [1–6]). For example, if q � p is an odd prime, Cao andWang [5] proved the following interesting conclusion: let p be an odd prime with p ≡ 1mod 3. If 3 is not a cubic residue modulo p, then we have the identity


Introduction
For any integer q ≥ 3 and integer m, the classical cubic Gauss sum A(m, q) is defined as follows: where as usual, e(x) � e 2πix and i 2 � − 1.
is sum plays a very important role in the study of the elementary number theory and analytic number theory, so there are many people who had studied the arithmetical properties of A(m, q) and related contents (see [1][2][3][4][5][6]). For example, if q � p is an odd prime, Cao and Wang [5] proved the following interesting conclusion: 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 where 4p � d 2 + 27 · b 2 and d is uniquely determined as follows: d ≡ 1 mod 3 On the other hand, we define the Dedekind sums S(h, q) as follows.
Let q be a natural number and h be an integer prime to q. e classical Dedekind sum S(h, q) is defined as where and give some exact computational formulae for (5) with q � p, an odd prime, where h and k are any two fixed positive integers. About this problem, so far no one seems to consider; at least, we have not seen any related papers before. Of course, this problem is meaningful, and it can describe the mean value distribution properties of the two different sums. It is clear that if (p − 1, 3) � 1, then from the properties of the complete residue system modulo p, we have A(m) � 0. is time (5) is meaningless. So, in the following, we only consider the case p ≡ 1 mod 3.
e main purpose of this paper is to prove the following several results. Theorem 1. Let p be a prime with p � 12h + 1. en, for any positive integers k and h with (2, h) � 1, we have the identity Theorem 2. Let p be a prime with p � 12h + 7. en, for any positive integer k, we have the identity where χ 2 is Legendre's symbol modulo p, λ denotes any third- satisfies the third-order linear recursive formula as follows: Theorem 3. Let p be a prime with p ≡ 1 mod 3. en, for any positive integer k, we have the asymptotic formula as follows: where exp(y) � e y and λ is a three-order character modulo p.
From eorem 3, we may immediately deduce the following two corollaries. Corollary 1. Let Let p be a prime with p ≡ 1 mod 3, then we have

Several Lemmas
To complete the proofs of our all theorems, we need to prove several simple lemmas. Hereinafter, we shall use some properties of the character sums and Gauss sums, and all of these contents can be found in [15], so they will not be repeated here.

Lemma 1.
Let p be an odd prime with p ≡ 1 mod 3 and λ be any third-order character modulo p, then we have the identity where τ(λ) � p− 1 a�1 λ(a)e(a/p) denotes the classical Gauss sums, and 4p � d 2 + 27 · b 2 , where d is uniquely determined by d ≡ 1 mod 3.

Proof.
e proof of this lemma is shown in the study by Zhang and Hu [2] or Berndt and Evans [16].

Lemma 2. Let p be an odd prime with p ≡ 1 mod 3. en, for any integer k ≥ 3 and
we have the third-order linear recursive formula as follows: with the initial values H(0, p) Proof. Let λ be any third-order character modulo p; then, for any integer 1 ≤ m ≤ p − 1, from the properties of the third-order characters and the classical Gauss sums modulo p, we have the identity Note that τ(λ)τ(λ) � p and λ 2 � λ; from (16), we have From (16) and Lemma 1, we also have From (17) and the properties of the character sums modulo p, we have Combining (18) and (19), we can deduce that Similarly, from (16), (18), and (21), we have Journal of Mathematics 3 If k ≥ 3, then from (18), we have or Now, Lemma 2 follows from (21)-(25) and H(0, p) � p − 1.
□ Lemma 3. Let p be a prime with p ≡ 1 mod 3. For any positive integer k and any character χ mod p, if χ 3 ≠ χ 0 , then we have the identity where χ 0 denotes the principal character modulo p.
Proof. Let g be a primitive root modulo p. en, from the definition of A(m, p) and the properties of the complete residue system modulo p, we have From (27), we have If χ 3 ≠ χ 0 , then χ 3 (g) ≠ 1. So, from (29), we have the identity is proves Lemma 3.