Topics in Number Theory

这个奖项是关于数论的问题，特别是l -函数的理论。数论的一个中心主题是了解素数的性质和分布。通常，通过将相关信息打包到称为l函数的分析对象中来理解这类问题。这些l -函数的研究是数论中的一个经典话题，始于19世纪黎曼和狄利克雷的工作，但仍然是现代数学的中心焦点，黎曼假设和伯奇&斯温纳顿-戴尔猜想等问题是该主题的核心。这一领域的进展有时在密码学和理论计算机科学中得到应用。研究者将继续在分析、数论和组合学的界面上进行他的工作。特别是，他将继续研究乘法数论（例如随机乘法函数的分布、Weyl和的平均值和奇异级数的平均值），并理解l函数中心值的矩和值分布。这笔拨款还将用于帮助培训该地区的研究生。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 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.