Topics in Analytic Number Theory and Additive Combinatorics

该项目涉及解析数论和加性组合数学领域的研究。这些领域中的一类重要问题涉及素数的分布，这是一个在理论计算机科学、密码学和金融安全中有重要应用的主题。这个研究项目的主要目标之一是通过进一步发展经典的分析工具和更现代的加法组合工具，以及了解如何将这两种不同的工具有效地结合在一起，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，满足解析数论应用的更强结果的需要将推动加法组合学的发展，反之亦然。更具体地说，PI将通过开发和组合前面提到的分析和组合工具来继续他围绕涉及素数的加法问题的工作。利用L函数理论和筛子理论的经典方法已经在这方面取得了很大的成果，人们期望通过结合加性组合数学的工具可以超越经典方法所能达到的效果。PI还将继续研究乘法函数和von-Mangoldt函数的Gowers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Gowers规范的研究是应用加法组合工具的基本分析输入。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.