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.

解析数论中各类和式的均值问题在数论研究中发挥着重要的作用,许多著名的数论难题都与之密切相关。本项目利用Gauss和的性质研究关于指数和与特征和的混合均值问题，在研究过程中，深入分析了解析数论研究中涉及到的各类方法，通过特征和与Gauss和的性质给出了一系列渐近公式与恒等式。主要内容包括：研究了广义二次Gauss和及特征和的2k次幂的均值，并给出了较强的渐近公式；讨论了一类特殊二项指数和的四次均值，得到了精确的计算公式，并给出了包含一类特殊二项指数和与特征和的恒等式；研究了著名数学家Erdős提出的一类丢番图方程，并在Sierpiński研究的基础上对形式相近的一种丢番图方程，并得到了其正整数解；研究了一些特殊整数集上三次剩余的计算问题，通过三阶特征与经典高斯和的性质给出了计数函数的一些精确公式。

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