The hybrid power mean of quartic Gauss sums and Kloosterman sums

Abstract The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind fourth hybrid power mean of the quartic Gauss sums and Kloosterman sums, and give an exact computational formula for it.


Introduction
Let q 3 be a positive integer. For any positive integer k 2, integers m and n, the k-th Gauss sums G.m; kI q/ and Kloosterman sums K.m; nI q/ are defined as G.m; kI q/ D where P 0 q aD1 denotes the summation over all 1 Ä a Ä q such that .a; q/ D 1, e.y/ D e 2 iy , and a denotes the multiplicative inverse of a mod q (aa Á 1 mod q).
Concerning the various properties of G.m; kI q/ and K.m; nI q/, many authors have studied them, and obtained several results, see [1][2][3][4][5][6][7]. For example, from the A.Weil's important work [1], one can get the upper bound estimatě where p is an odd prime, denotes any Dirichlet character modp, and k denotes the big-O constant depending on k.
Zhang Wenpeng and Liu Huaning [2] studied the fourth power mean of G.m; k; p/, and obtained some sharp asymptotic formulae for it.
T. Estermann [3] proved the upper bound jK.m; nI q/j Ä .m; n; q/ where .m; n; q/ denotes the greatest common divisor of integers m; n and q, d.q/ denotes the number of divisors of q. For general odd integer q 3 and .n; q/ D 1, Zhang Wenpeng [8] proved the identity where .q/ is Euler function, !.q/ denotes all distinct prime divisors of q, Q pkq denotes the product of all prime divisors of q such that p j q and .p; q=p/ D 1.
Some related works can also be found in [8][9][10]. Now let p be an odd prime with p Á 1 mod 4. For any positive integer k, we consider the fourth hybrid power mean In this paper, we are concerned with the calculating problem of (1). Regarding this, as far as we knew, it seems that nobody has studied it yet, at least we are not aware of such work. The problem is interesting, because it can help us to understand more accurate information of the hybrid mean value of the quartic Gauss sums and the classical Kloosterman sums.
In this paper, we shall use the analytic methods and the properties of Gauss sums to study the calculating problem of (1), and give an interesting computational formula for (1) with h D 1. That is, we shall prove the following: Theorem. Let p be an odd prime with p Á 1 mod 4. Then we have the identity where 4 is any four order character mod p, . / D P p 1 aD1 .a/e a p Á denotes the classical Gauss sums.
Note that j . 4 /j D p p, from our theorem we may immediately deduce the following two corollaries: Let p be an odd prime with p Á 5 mod 8. Then we have the asymptotic formula Corollary 1.2. Let p be an odd prime with p Á 1 mod 8. Then we have the asymptotic formula In this section, we give several lemmas which are necessary for the proof of our theorem. Hereinafter, we shall use many properties of Gauss sums, all of them can be found in [11], so we will not be repeated here. First we have the following: Lemma 2.1. Let p be an odd prime with p Á 1 mod 4, be any character mod p. Then we have the identity where 4 is any four order character mod p, p Á D 2 is the Legendre's symbol, and . / D where we have used the identity P p 1 aD1 .a/e ma p Á D .m/ . /. Now for any character mod p, note that the identity This proves Lemma 2.1.