Topics in Number Theory

该奖项涉及数论问题，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，这些问题可以通过将相关信息打包到称为L函数的分析对象中来理解。 对这些L-函数的研究是数论中的一个经典话题，始于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，像黎曼假设和伯奇-斯温纳顿-戴尔定理等问题是这一主题的核心。 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。调查员将继续他的工作在接口的分析，数论和组合学的主题。 特别是，他将继续研究乘法数论（例如在分布的随机乘法函数，平均Weyl总和，平均奇异系列），并在了解的时刻和价值分布的中心价值的L-功能。 该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.