Topics in Number Theory

该奖项涉及数论中的问题，特别是L函数理论。数论的一个中心主题是理解素数的性质和分布。通常，这样的问题是通过将相关信息打包成称为L函数的分析对象来理解的。对这些L函数的研究是数论中的一个经典课题，从19世纪黎曼和狄里克莱特的工作开始，但仍然是现代数学的中心焦点，诸如黎曼假设和Birch&Amp；Swinnerton-Dyer猜想等问题是该主题的核心。这一领域的进展有时会在密码学和理论计算机科学中得到应用。研究人员将在分析、数论和组合学的界面上继续他的工作。特别是，他将继续研究乘法数论(例如，随机可乘函数的分布、Weyl和的平均、奇异级数的平均)，并了解L函数中心值的矩和值分布。这笔奖金还将用于帮助培训该领域的研究生。这一奖项反映了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 award is concerned with problems in number theory, specifically the theory of L-functions. One central theme in number theory is to understand the properties and distribution of prime numbers. Often such problems are understood by packaging the relevant information into analytic objects known as L-functions. The study of these L-functions is a classical topic in number theory beginning with the work of Riemann and Dirichlet in the nineteenth century, but yet continues to be a central focus of modern mathematics, with problems like the Riemann Hypothesis and the Birch & Swinnerton-Dyer conjectures lying at the heart of the subject. Progress in this area has sometimes found applications in cryptography and theoretical computer science. The investigator will continue his work on topics at the interface of analysis, number theory, and combinatorics. In particular he will continue research in multiplicative number theory (for example in the distribution of random multiplicative functions, averages of Weyl sums, and averages of singular series), and in understanding the moments and value distribution of central values of L-functions. The grant will also be used to help train graduate students in the area.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.