Topics in Number Theory

该项目涉及分析数论领域的研究。 在这一领域的一类重要问题涉及L-函数，它编码算术信息。 L函数的一个例子是黎曼zeta函数，它控制素数的分布。 研究者的主要目标之一是在L-函数的值分布和零点方面取得进展。 同时，人们希望使用这些信息来提取应用程序的算术问题。 虽然这些主题在数学中具有内在的兴趣，但数论的进展在密码学和理论计算机科学中有重要的应用。 更具体地说，研究人员将继续他的调查时刻和价值分布的L-功能，特别是他最近的工作与Radziwill。调查员还将与Granville、哈珀和Koukoulopoulos合作，继续研究乘法函数。 除了这两个主要项目外，研究人员还将与合作者一起研究分析，数论和组合学界面上的各种问题，并继续他在这一领域培养研究生的工作。

This project involves research in the area of analytic number theory. An important class of problems in this area concerns L-functions, which encode arithmetic information. One example of an L-function is the Riemann zeta function, which controls the distribution of prime numbers. One of the investigator's main goals is to make progress on the value distribution and zeros of L-functions. Concurrently, one hopes to use such information to extract applications to problems in arithmetic. While these topics are of intrinsic interest in mathematics, progress in number theory has had important applications in cryptography and theoretical computer science. More specifically, the investigator will continue his investigations on moments and value distribution of L-functions, extending in particular his recent work with Radziwill. The investigator will also continue work on multiplicative functions, in collaboration with Granville, Harper, and Koukoulopoulos. Besides these two main projects, the investigator will work with coauthors on various problems at the interface of analysis, number theory, and combinatorics, and continue his work in training graduate students in this area.