Topics in Analytic Number Theory and Additive Combinatorics

该项目涉及分析数论和加性组合学领域的研究。这些领域中一类重要的问题涉及素数的分布，这个主题在理论计算机科学、密码学和金融安全中有着重要的应用。本研究项目的主要目标之一是通过进一步发展经典分析工具和更现代的加法组合工具，并了解如何有效地将这两种不同的工具结合在一起，在素数和其他相关算术对象的分布性质方面取得进展。人们希望，为了满足分析数论应用的需要，需要更强的结果，这将推动加性组合学的发展，反之亦然。更具体地说，PI将通过开发和结合前面提到的分析和组合工具，继续围绕涉及素数的可加性问题开展工作。使用l函数理论和筛理论的经典方法在这方面取得了很大的成果，并且期望通过结合加性组合学的工具可以超越经典方法所能达到的目标。PI还将继续研究乘法函数和冯-曼戈特函数的高尔斯范数理论。这些是涉及这些函数的经典指数和估计的“高阶”扩展。对这些函数的一般高尔斯范数的研究是应用加性组合工具必不可少的分析输入。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project involves research in the areas of analytic number theory and additive combinatorics. An important class of problems in these areas concern the distribution of prime numbers, a topic that has had important applications in theoretical computer science, cryptography, and financial security. One of the main goals of this research project is to make progress on the distributional properties of primes and other related arithmetic objects, by developing further the classical analytic tools and the more modern additive combinatorial tools, and by understanding how to effectively combine these two different tools together. It is hoped that the need for stronger results to cater for applications in analytic number theory will drive the development of additive combinatorics, and vice versa. More specifically, the PI will continue his work surrounding additive problems involving primes by developing and combining aforementioned analytic and combinatorial tools. The classical approaches using the theory of L-functions and sieve theory have achieved great results in this direction, and it is expected that by incorporating tools from additive combinatorics one can go beyond what the classical methods could achieve. The PI will also continue to investigate the theory of Gowers norms for multiplicative functions and for the von-Mangoldt function. These are "higher-order" extensions to classical exponential sum estimates involving these functions. The study of general Gowers norms for these functions is an essential analytic input for applying additive combinatorial tools.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.