Analytic Theory of L-functions and Modular Forms

本文的主要内容涉及L-函数的均值、零点分布和素数分布之间的关系。 研究者和他的同事们研究L函数的均值乘以一个“缓和因子”，以证明均值和随机矩阵理论预测的L函数的零点分布之间的等价性。 研究者和他的同事们还研究了零点分布与短间隔内素数分布变化之间的联系。 在一个单独的项目中，研究人员开发了计算机程序来构建Hecke同余子群的Maass形式。 由此产生的数据证实了随机矩阵理论的预测。这项工作的动机是最近发现的数论和物理学之间的联系。 这些联系揭示了经典数论（数学）和量子混沌（物理）之间的关系。 用来展示这种联系的主要工具是随机矩阵理论，这是一个作为解释核物理中某些实验结果而出现的学科。 随机矩阵理论已经被发现与其他物理现象有关，以及纯理论数学的几个领域。 通过利用随机矩阵理论的这种联系，可以使用数学来解释物理实验中观察到的现象，并使用潜在的物理学作为发现新数学的动机。 这样，通过结合这两种办法，就有可能取得比单独取得的进展更大的进展。 该补助金中的大多数项目将在本科研究生的协助下进行。

The main part in this proposal concerns the relationship between mean-values of L-functions, the distribution of zeros of L-functions, and the distribution of the prime numbers. The investigator and his colleagues study mean-values of L-functions times a "mollifier," in order to prove an equivalence between the mean-value and the distribution of zeros of the L-function predicted by random matrix theory. The investigator and his colleagues also study the connection between the distribution of zeros and variations in the distribution of primes in short intervals. In a separate project the investigator develops computer programs to construct Maass forms on Hecke congruence subgroups. The resulting data confirms the predictions of random matrix theory.The work in this proposal is motivated by recently discovered connections between number theory and physics. These connections reveal a relationship between classical number theory (mathematics) and quantum chaos (physics). The main tool used to exhibit the connection is Random Matrix Theory, a subject which arose as an explanation for certain experimental results in nuclear physics. Random Matrix Theory has since been found to be related to other physical phenomena, as well as several areas of purely theoretical mathematics. By exploiting this connection with Random Matrix Theory it is possible to use mathematics to explain phenomena observed in physical experiments, and to use the underlying physics as motivation for discovering new mathematics. In this way, more progress is possible by combining the two approaches than could be achieved separately. The majority of projects in this grant will be carried out with the assistance of undergraduate research students.