Topics in Analytic Number Theory

申请者将继续他在乘法数论和l -函数解析理论方面的研究。本研究的长期目标是阐明l函数的零的大小和分布，并探索这些结果对算术问题的影响。最近，作者获得了一大类l函数的“弱次凸性”结果，并将研究该结果的推广和推广。与Conrey和Iwaniec一起，提出了一个渐近大筛，有望在某些族的l -函数的零点和矩上给出新的结果。他还将研究与模形式的等分布有关的问题。在过去的十年里，他与Andrew Granville共同研究了乘法函数的平均值，并发现了许多引人注目的应用。提出者将继续这些研究，并与格兰维尔一起从这一观点对解析数论中的许多主题进行统一处理。提案人在分析数论领域涉猎广泛。l -函数在这一领域中占有重要地位，它编码了大量的算术信息。一个关键的例子是黎曼ζ函数，它包含了许多关于素数分布的信息，而对它的零的理解是数学中的基本问题之一。提案人的工作动机是想要理解这种l函数的行为。数论中的问题是数学家们固有的兴趣，并提出了艰巨的挑战。此外，数论的进步在密码学和理论计算机科学中也有应用。

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.