摘要
Let α be a real number, x≥A≥2, e(θ) = e^(2xiθ), and suppose Λ(n) is Mangoldt's func-tion. In this paper the following result is mainly proved: Let ε be an arbitrarily small po-sitive number, and x^(91/(96+ε))≤A≤x. Then for any given positive c, there exists a positive c_1such that for A^(-1)log^cx ≤|α| ≤(logx)^(-c_1) there exitss sum from x-A<n≤x (Λ(n)e(nα)? A(logx)^(-c).
Let α be a real number, x≥A≥2, e（θ） = e<sup>2xiθ</sup>, and suppose Λ（n） is Mangoldt’s func-tion. In this paper the following result is mainly proved: Let ε be an arbitrarily small po-sitive number, and x<sup>91/（96+ε）</sup>≤A≤x. Then for any given positive c, there exists a positive c<sub>1</sub>such that for A<sup>-1</sup>log<sup>c</sup>x ≤|α| ≤（logx）<sup>-c<sub>1</sub></sup> there exitss sum from x-A<n≤x (Λ（n）e（nα）? A（logx）<sup>-c</sup>.
基金
Project supported by the National Natural Science Foundation of China.
