Bounds for certain exponential sums
Bounds for certain exponential sums
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DOI：
10.4310/ajm.2000.v4.n4.a3
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发表时间：
2000
影响因子：
0.6
通讯作者：
Todd Cochrane;Zhiyong Zheng
中科院分区：
文献类型：
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作者：
Todd Cochrane;Zhiyong Zheng
其中p是素数幂，χ mod p是Dirichlet特征标，a，B，n是n ≥ 2的整数。第一个和研究与华林的问题，我们有一个经典的结果，由于教授华[10]。据作者所知，第二个和以前没有研究过。我们希望它能用于推广Waring问题的工作中。Davenport和海尔布龙在[6]中证明了（1.3）S（ax + bx，p）np（B，p），如果p a，其中当n = 3时θ = 2/3，当n ≥ 3时θ = 3/4. Hua [10]证明了对于所有n ≥ 2，θ = 1/2（也参见Vaughan [19]的引理4.1）。Hua的证明依赖于有限域上指数和的Weil估计（参见Buberi [1]或施密特[15]）。在这种情况下，我们有（1.4）|S（ax + bx，p）|≤（n− 1）p 2，如果p a.对于m ≥ 2，根据洛克斯顿和Smith [13]以及Smith [17]的工作，我们得到（1.5）|S（ax + bx，p）|≤ dn−1（p）p m 2（n，p）1 2，其中dn−1（p）是p作为n−1个正整数的乘积的表示数，n是多项式ax+bx的导数的判别式。经过洛克斯顿和Vaughan的改进，[14]的定理1意味着下面的估计（1.6）|S（ax + bx，p）|≤（n− 1）pm +τ0 2（D，p）1 2，其中D是多项式ax + bx的导数的差，如果p ≤ n，τ0 = 1，如果p > n，τ0 = 0。最近，Dabrowski和Fisher为这类指数和建立了更好的界。在限制条件p ab，pn和
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