三次华林-哥德巴赫问题的定量研究

In this project, we will do some researches about the conjecture that an integer satisfying some necessary congruence conditions can be represented as the sum of four prime cubes in the cubic Waring-Goldbach problem. By known results, the newest progresses and our own mean value results built on the prime variables and by restricting the integer variable and prime variables separately or simultaneously lying in the short intervals, we hope to obtain a series of approximations to this conjecture. We will study the following problems in focus: (1) For the known result of an integer that is the sum of four cube primes in short intervals, we continue to get the qualitative improvements about the length of the short intervals. (2) Use the known result of asymptotic formula in short intervals about the mean value of the number of representation to build the new and needed result in order to get quantitative estimation on the positive density.Then we improve the value of density. (3) If at most two variables are almost primes, we consider the nontrivial exception set and the variables restricted in short intervals, and try to obtain the quantitative relations between exceptional sets and short intervals.Throughout the proof in this project, we will make full use of the crucial technique of enlarging major arcs in circle method, and take into account of dealing with the minor arcs. By making a better combination with the sieve method, we will take different forms of sieve functions in order to get the corresponding result of short intervals. We will pay special attention to the combination and innovation of the method that dealing with the minor arcs at the.same time, the series of improvements are the descriptive and approximation to the conjecture.

本项目拟对三次华林-哥德巴赫问题中满足必要同余条件整数可写成四个素变量立方和猜想展开研究.拟利用素变数指数和已有结果、最新进展和建立新均值结果，通过限制整变量和素变量分别或全部落在小区间上对此猜想展开逼近.主要考查:(1)针对表为四素数立方和整数在小区间上已有结果，期望对小区间的长度估计获得定量改进.(2)利用小区间上表法个数的均值估计渐近公式，对正密度的数值大小获得定量估计，改进现有的密度数值.(3)当素变量至多两个为殆素数时，考查非平凡例外集的同时将各变量限制到小区间上,尝试建立例外集和小区间的定量关系式.研究过程中充分利用圆法中扩大主区间的关键技术兼顾对余区间处理.同时与筛法紧密结合,通过加入不同形式的筛函数，建立相应的小区间结论.证明过程中特别注意对余区间处理方法综合运用及创新,得到的定量改进结果将是对猜想的刻画.

华林-哥德巴赫问题是堆垒数论中一个重要的研究课题，研究某个满足必要同余条件的正整数的可表性质，对于这类问题，我们通常用经典的解析方法和工具展开研究。本项目主要对数论中的三次华林-哥德巴赫问题展开定量研究，研究可表整数的正密度数值的定量估计，改进现有的密度数值。此外，还对定义在全体正整数上的数论函数的均值估计展开定性研究，并考虑一些算术应用。某种意义下数论中的三次华林-哥德巴赫问题中的四素数立方和猜想可对照偶数的哥德巴赫猜想，考查小区间内其正密度问题可视为从其他角度出发来逼近猜想。数论函数算术平均值的性质比数论函数本身要规律很多，用其均值意义下的结论来推测数论函数的性质，无疑也是有意义的。