Topics in Analytic Number Theory

提出者将继续他的调查在乘法数论和解析理论的L-函数。 本研究的长期目标是阐明L-函数零点的大小和分布，并探索这些结果对算术问题的影响。 最近的提议者获得了一个``弱次凸性"的结果为一大类的L-功能，他将调查的推广和扩展的结果。 与Conrey和Iwaniec一起，提议者正在开发一种渐近大筛，该筛有望在某些家族中给出关于L-函数的零点和矩的新结果。 他还将研究与模块形式的均匀分布有关的问题。 在过去的十年里，他与安德鲁·格兰维尔一起研究了乘法函数的平均值，这项研究发现了许多引人注目的应用。 提议者将继续这些调查，并与格兰维尔正在努力统一处理许多议题的解析数论从这个角度来看。 这个提议者广泛地在解析数论领域工作。L-函数在这一领域具有核心重要性，并编码了大量的算术信息。 一个关键的例子是黎曼zeta函数，它包含了很多关于素数分布的信息，而理解它的零点是数学中的基本问题之一。 提议者的工作动机是希望了解这样的L-函数的行为。 数论中的问题是数学家的内在兴趣，并提出了艰巨的挑战。 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.