Topics in Number Theory

这个项目涉及解析数论领域的研究。这一领域中的一类重要问题涉及编码算术信息的L函数。L函数的一个例子是Riemann 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，人们希望利用这些信息来提取算术问题的应用。虽然这些主题在数学中具有内在的兴趣，但数论的进步在密码学和理论计算机科学中有重要的应用。更具体地说，调查者将继续他对L函数的矩和值分布的研究，特别是他最近与Radziwill的工作。调查员还将与Granville、Harper和Koukoulopoulos合作，继续研究乘性函数。除了这两个主要项目外，研究人员将在分析、数论和组合学的界面上与合著者合作解决各种问题，并继续他在这一领域的研究生培训工作。

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.