素数堆垒问题中的圆法、筛法和Linnik方差法

The research on the theory of additive problem of prime numbers is an interdisciplinary study field, which covers two of the most important fields in number theory, the distribution of primes and Diophantine equations. Therefore, The research on this topic is of great theoretical significance. .This research includes: (1) We will continue to pay attention to classical analytic methods and classical problems, especially the important role of circle method and sieve methods to the theory of additive problem of prime numbers. We will research Waring-Goldbach problem for unlike powers systematically, and give some general results. We will look for more Goldbach-Linnik type partitions. (2) The Linnik dispersion method will be studied in depth, which makes it a new method to study the theory of additive problem of prime numbers. This will lead to get some important progress of the final proof of the four-prime-squares-conjecture. (3) We will discuss the structure of prime solutions of general Diophantine equations with degree three or more, and give the proof of some special cases of Sarnak conjecture on additive problem of prime numbers, which provides more evidences for Sarnak conjecture..Through this project, we plan to combine the new technology of Linnik dispersion method, with classical analytic methods in order to get some influential original results in the field of additive problem of prime numbers..It should be pointed out that there is a close relation between this project and the Sarnak conjecture which has been attracting increasing attentions recently. Therefore, this project is a popular research standing at the frontier of analytic number theory.

素数堆垒问题是丢番图方程和素数分布这两个数论重要研究领域的交叉领域，具有重要的理论意义。.本项目的主要研究内容是：一、继续关注经典解析方法和经典问题，尤其是圆法和筛法对素数堆垒问题的重要作用，系统研究不等次幂华林-哥德巴赫问题，给出一般性的结果；寻找更多的Goldbach-Linnik型分拆式；二、深入研究Linnik方差法，使其成为研究素数堆垒问题的新方法，为四素数平方和猜想的最终证明提供阶段性结果；三、探讨三次及以上一般丢番图方程素数解的结构，证明Sarnak猜想在素数堆垒问题上的特例，为Sarnak猜想提供更多实证。.计划通过本项目的实施，将Linnik方差法的新技术与经典解析方法相结合，在素数堆垒问题研究领域取得有影响的原创性成果。.本研究课题与已引起普遍关注的Sarnak猜想研究具有密切的联系，属于解析数论领域的热门课题和前沿课题。

堆垒素数论是丢番图方程和素数分布这两个数论重要研究领域的交叉领域，具有较强的理论意义，属于解析数论领域的热门课题和前沿课题。.本项目的研究内容包括：一、利用圆法和筛法的最新工具和最先进的技术研究一般的不等次幂华林-哥德巴赫问题和Goldbach-Linnik型分拆式；二、利用Linnik方差法结合圆法和筛法研究二次华林哥德巴赫问题；三、进一步深入研究一般素数堆垒问题的解的结构，得到三次及以上的华林-哥德巴赫问题的深刻结果。.本项目的研究成果包括：一、首次定量得到不等次幂华林哥德巴赫问题的Baker常数；大幅改进了三次及以上的一般华林哥德巴赫问题的Baker常数；二、大幅改进了四素数的平方与2的方幂之和的方程组问题中的常数；三、大幅改进了四次和五次华林哥德巴赫问题的例外集；四、首次定量得到斐波那契数的Romanoff问题的密度。.本项目利用和发展了圆法和筛法等经典解析数论方法，并结合Linnik方差法，建立新的结果或改进已有结果。.本项目在堆垒素数论研究领域取得了有一定学术影响力的原创性成果。

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