几个堆垒素数问题定量研究

The purpose of this project is to give some quantitative studies related to the following three kinds of problems in the field of additive prime number theory. (1)Consider to remove the "sufficiently large"condition in the results of expressing integers as sums of lower prime powers,such as that in the classical theorem of professor Hua that every "sufficiently large" integer congruent to 5 modulo 24 is a sum of five prime squares.The investigation of these kinds of problems is interesting and valuable if one notes and compares it with a recent important result of Helfgott,which gave a complete solution to the classical odd Goldbach conjecture. (2)Using the latest numerical results in the field of arithmetic combinatorics and zeros of Dirichlet L functions, we continue to consider the numerical upper bound of the least prime solution of some Linnik-Gallagher equations, and consider the improvement of Baker's constant on the upper bound of solution of some equations with prime variables. (3)By some delicate investigation of the ideas and the methods of Yitang Zhang and James Maynard,we manage to discuss some related new kinds of mean value theorems,and then by considering the sieve of Goldston-Pints-Yildirim, we come to give some new quantitative results on the related problems,such as the prime twin conjecture itself.

本项目主要围绕下述三类堆垒数论问题开展定量研究：（1）借鉴Helfgott近期完全"移除"三素数定理中"充分大"条件，从而完整解决奇数Goldbach猜想的思想，寻求移除一些非线性低方幂素变数方程可解性对于"充分大"条件的依赖，比如移除华罗庚先生五平方素数定理中的"充分大"条件等。（2）结合近期算术组合和零点分布的有关研究及进展，继续研究Dirichlet L函数零点分布和密度估计的定量估计，进而考虑Linnik-Gallagher型方程最小素数解的定量上界估计，并继续考虑Baker关于素变数方程解的定量上界估计。（3）结合张益唐对孪生素数猜想的重大推进及Maynard等人的相关跟进，考虑有关均值定理的推广改进和应用，进而通过细化Maynard对Goldston-Pints-Yildirim型筛法的推广，探讨某些相关定量结果的直接改进和有关延伸。

本项目主要围绕以下几类堆垒素数问题开展定量研究：移除了一些重要非线性低方幂素变数方程可解性对于充分大条件的依赖问题、研究了几类Linnik-Gallagher型方程最小素数解的定量估计问题、得到了一些素数分布均值估计的定量结果。经过四年的研究，在主持人王天泽的协调下，李伟平、王天芹、赵峰、戈文旭、范云艳、陈国华、韩延婷按照研究计划，比较顺利的开展了相关研究工作。项目组成员共发表论文17篇，其中SCI论文11篇，中文核心论文4篇。指导硕士生发表论文十余篇，其中SCI论文1篇。获河南省自然科学优秀学术论文二等奖一项，河南省教育厅科技成果一等奖一项。在Waring-Goldbach问题以及由此引出的素变数丢番图逼近问题方面取得了一系列创造性的成果。在素变数指数和的定量估计，算术数列中的素数分布的定量估计方面取得了一些突破，得到了一些创新性的成果。在Linnik-Gallagher型方程最小素数解方面，结合近几年数论的新方法，发挥自己在这方面的研究优势，解决了几类方程的最小素数解的估计问题。