On exponential sums over primes and application in Waring-Goldbach problem

作     者：REN Xiumin 

作者机构：Department of Mathematics Shandong University Jinan 250100 China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2005年第48卷第6期

页      面：785-797页

核心收录：

学科分类：07[理学] 0701[理学-数学] 

基　　金：The author is supported by Post-Doctoral Fellowsbip of The University of Hong Kong. 

主　　题：exponential sums over primes, Waring-Goldbach problem, circle method. 

摘      要：In this paper, we prove the following estimateon exponential sums over primes: Let κ≥ 1, βκ = 1/2 + log κ/log 2, x ≥ 2 and α = a/q + λ subject to (a, q) = 1, 1 ≤ a ≤ q,and λ R. ThenΣxm≤2x∧(m)e(αmk)(d(q))βk(logx)c(x1/2√q(1+|λ|xk)+x4/5+x/√q(1+|λ|xk)).As an application, we prove that with at most O(N7/8+ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.