One Kind Special Gauss Sums and their Mean Square Values

In this paper, we introduce one kind special Gauss sums; then, using the elementary and analytic methods to study the mean value properties of these kind sums, we obtain several exact calculating formulae for them.


Introduction
Let q and k be two positive integers with q ≥ 3. For any integer a ≥ 1 and Dirichlet character χmodq, we write where [x] denotes the greatest integer less than or equal to x. en, we define the summations G(n, χ, k; q) and H(n, χ, k; q) as G(n, χ, k; q) � q k − 1 a�1 χ R k (a) e na q , where e(y) � e 2πiy and n is any integer. It is clear that G(n, χ, k; q) is a generalization of the classical Gauss sums. In fact, if k � 1, then we have R 1 (a) � a and at is, G(n, χ, 1; q) becomes the classical Gauss sums G(n, χ; q). Gauss sums play a very important role in the study of analytic number theory, and many number theory problems are closely related to it, so some scholars have studied the properties of the classical Gauss sums and obtained many meaningful and interesting results; some of them can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. For this reason, we think that it is necessary and meaningful to study the properties of G(n, χ, k; q).
On the contrary, H(n, χ, 2; q) is related to Dirichlet L-function L (1, χ). In fact, if p is an odd prime, χ(− 1) � − 1 and n � 0, then we have erefore, the study of the sums G(n, χ, n; q) and H(n, χ, n; q) has extensive theoretical significance and application values.
In this study, as an attempt in this direction, we first study the mean value properties of G(n, χ, k; q) and H(n, χ, k; q). We shall use the elementary and analytic methods to prove the following several results. Theorem 1. Let p be an odd prime. en, for any integer n with (n, p) � 1, we have the identity where a denotes the solution of the congruence equation ax ≡ 1modp.
Theorem 3. Let p be an odd prime. en, for any nonprincipal character χmodp, we have the identities From eorems 1 and 2, we can also deduce the following corollaries: Corollary 1. Let p be an odd prime. en, we have the asymptotic formula χmodp Corollary 2. Let p be an odd prime. en, we have the asymptotic formula χmodp Notes. In fact, for any integer h ≥ 4, using our methods and the results in [15], we can give an exact calculating formula for the general 2hth power mean: However, when h is large, the calculation is more complicated, so we do not consider it.

Several Lemmas
is section, we need to give some simple lemmas, which are necessary in the proofs of our theorems. Of course, the proofs of these lemmas also need some knowledge of elementary and analytic number theory, in particular, the properties of the character sums and the classical Gauss sums modulo p. All these can be found in [16,17], we do not repeat them. First, we have the following.

Lemma 1. Let p be an odd prime and k ≥ 2 be a fixed integer. en, for any nonprincipal Dirichlet character χmodp, we have the identity
where τ(χ) � is number pair (m, i) not only exists, but it is unique.
is proves the first formula in Lemma 2. Similarly, note that the identity