Topics in Analytic Number Theory

作者将继续研究乘法数论和L函数的解析理论。这项研究的长期目标是阐明L函数的零点的大小和分布，并探索这些结果对算术问题的影响。最近，作者得到了一大类L函数的一个“弱次凸性”结果，他将研究该结果的推广和推广。与Conrey和Iwaniec一起，作者正在开发一个渐近大筛子，它有望给出某些族中L函数的零点和矩的新结果。他还将研究与模形式的均匀分布有关的问题。作者与安德鲁·格兰维尔合作，在过去的十年里一直在研究乘法函数的平均值，这项活动已经发现了一些引人注目的应用。提出者将继续这些研究，并与格兰维尔一起，从这个角度统一处理解析数论中的许多主题。该提出者在解析数论领域有广泛的工作。L函数是这一领域的核心，它编码了大量的算术信息。一个关键的例子是Riemann Zeta函数，它包含了大量关于素数分布的信息，而对其零点的理解是数学中的基本问题之一。提出者的工作动机是希望了解这些L函数的行为。数论中的问题是数学家与生俱来的兴趣，并提出了巨大的挑战。此外，数论的进步已经在密码学和理论计算机科学中得到了应用。

The proposer will continue his investigations in multiplicative number theory and the analytic theory of L-functions. The long term goal of this research is to shed light on the sizes and distribution of zeros of L-functions, and to explore the consequences of such results for arithmetical problems. Recently the proposer obtained a ``weak subconvexity" result for a large class of L-functions, and he will investigate generalizations and extensions of that result. Together with Conrey and Iwaniec, the proposer is developing an asymptotic large sieve which promises to give new results on zeros and moments of L-functions in certain families. He will also investigate problems relating to the equidistribution of modular forms. Jointly with Andrew Granville, the proposer has been studying mean values of multiplicative functions over the last ten years, and this activity has found a number of striking applications. The proposer will continue these investigations, and together with Granville is working on a unified treatment of many topics in analytic number theory from this viewpoint. The proposer works broadly in the area of analytic number theory. L-functions are of central importance in this area, and encode a great deal of arithmetic information. A key example is the Riemann zeta function which contains much information about the distribution of prime numbers, and an understanding of its zeros is one of the fundamental problems in mathematics. The proposer's work is motivated by a desire to understand the behavior of such L-functions. The problems in number theory are of intrinsic interest to mathematicians, and pose formidable challenges. Moreover, progress in number theory has had applications in cryptography and theoretical computer science.