素变量混合幂丢番图逼近(Ⅱ)（英文）  

Diophantine Approximation With Prime Variables and Mixed Powers(Ⅱ)

作　　者：牟全武[1] 吕晓东[2,3] MU Quanwu LU Xiaodong(College of Science, Xi＇an Polytechnic University, Xi＇an, Shaanxi, 710048, P. R. China Department of Mathematics, Tongji University, Shanghai, 200092, P. R. China School of Math ematical Science, Yangzhou University, Yangzhou, Jiangsu, 225002, P. R. China)

机构地区：[1]西安工程大学理学院,西安陕西710048 [2]同济大学数学系,上海200092 [3]扬州大学数学科学学院,扬州江苏225002

出　　处：《数学进展》2017年第2期190-202,共13页Advances in Mathematics（China）

基　　金：Supported by NSFC(No.11201107,No.11271283,No.11501435);the Natural Science Basic Research Plan in Shaanxi Province of China(No.2016JM1013);the Starting Research Fund for Doctors of Xi'an Polytechnic University(No.BS1508)

摘　　要：设λ_1,λ_2,λ_3,λ_4是正实数,λ_1/λ_2是无理数和代数数,V是well-spaced序列,δ>0.证了:对于任意给定的大于或等于3的正整数k及任意ε>0,v∈V,v≤X,使得λ_1p_1~22+λ2p_2~2+λ_3p_3~3+λ_4p_4~k-v|<v^(-δ)没有素数解p_1,p_2,p_3,p_4的v的个数不超过O(X^(σ+2δ+ε)),这里σ满足:当3≤k≤4时σ=1-4/11k;当k≥5时,σ=1-2/11k.这改进了之前[Chinese Ann.Math.Ser.A,2015,36(3):303-312]的结果.Let λ1, λ2,λ3, λ4 be positive real numbers and suppose that λ1/λ2 is irrational and algebraic. Let v be a well-spaced sequence, δ〉 0. It is proved that for any given positive integer k≥ 3 and for any ε 〉 0, the number of v ∈V with v ≤ X for which ｜λ1p1^2 ＋ λ2p2^2 ＋ λ3p3^3＋ λ4p4^k- v｜〈 v ^-δ has no solution in primes p1,p2,pa,p4 does not exceed O（X^σ＋2δ＋ε）, where σ = 1- 4/11k for 3 ≤ k ≤ 4, and σ = 1-2/11k-16/11k^2（k＋1） for k ≥ 5. This gives an improvement of an earlier result in [Chinese Ann. Math. Ser. A. 2015. 36（3）： 301-312].

关 键 词：丢番图不等式 Davenport-Heilbronn方法 素数 

分 类 号：O156.4[理学—数学]