Bounds for certain exponential sums

Abstract

where p is a prime power , χ mod p is a Dirichlet character, a, b, n are integers with n ≥ 2. The first sum was studied in connection with Waring’s problem and we have a classical result due to professor Hua [10]. The second sum has not been studied before as far as the authors know. We hope it can be used in the work of generalizing Waring’s problem. In [6], Davenport and Heilbronn showed that (1.3) S(ax + bx, p) n p(b, p), if p a, where θ = 2/3 if n = 3, and θ = 3/4 if n ≥ 3. Hua [10] showed that θ = 1/2 for all n ≥ 2 (also see Lemma 4.1 of Vaughan [19]). Hua’s proof depends on Weil’s estimate for exponential sums over finite fields (see Bombieri [1] or Schmidt [15]). In this case we have (1.4) |S(ax + bx, p)| ≤ (n− 1)p 2 , if p a. For m ≥ 2, following the work of Loxton and Smith [13], and Smith [17], one has that (1.5) |S(ax + bx, p)| ≤ dn−1(p)p m 2 (∆, p) 1 2 , where dn−1(p) is the number of representations of p as a product of n−1 positive integers and ∆ is the discriminant of the derivative of the polynomial ax+bx. After an improvement by Loxton and Vaughan, Theorem 1 of [14] implies the following estimate (1.6) |S(ax + bx, p)| ≤ (n− 1)p m+τ0 2 (D, p) 1 2 , where D is the different of the derivative of the polynomial ax + bx, and τ0 = 1 if p ≤ n, and τ0 = 0 if p > n. Very recently, Dabrowski and Fisher established better bounds for exponential sums of this kind. Under the restriction p ab, p n, and