L-functions: Zeros and Values

研究者在随机矩阵理论的背景下研究L-函数的零点和值的统计性质。随机矩阵模型被用来猜想L函数矩的完全渐近展开式和解析连续性，并预测椭圆曲线的阶数的统计性质。一个C++L函数类库和一个前端应用程序也在开发中，用于计算L函数的零和值。调查人员正在使用这个软件来从数字上证实所做的预测，并将免费向公众发布，作为研究L函数的急需工具。数论中的许多问题都可以用所谓的L函数的性质来描述。这些函数编码了关于各种数论问题的深刻信息，在很大程度上仍然不屈服于数学分析。如果能详细地理解这些函数，数论中的许多深层次问题就会迎刃而解。令人惊讶的是，一个被称为随机矩阵理论的看似无关的领域，最初出现在与实验物理有关的学科，最近被数字理论家和物理学家发现，为模拟L函数的行为提供了一个框架。这种神秘的联系已经被成功地用来对L的行为做出迄今无法想象的预测--函数。这项提案中的工作涉及到探索这两个领域--数论和随机矩阵理论之间的联系。为了协助这个项目，调查者还准备了一个软件包，供公众免费使用，用于数值研究L函数。该奖项由代数、数论和组合数学计划、数值、符号和几何计算计划以及计算数学计划共同资助。

The investigator is studying the statistical properties of zeros and values ofL-functions in the context of random matrix theory. Random matrix models are being used to conjecture the full asymptotic expansion and analytic continuationfor the moments of L-functions, and to make predictions for the statistical behavior of ranks of elliptic curves. A C++ L-function class library along with a front end application is also being developed for computing zeros and values of L-functions. This software is being used by the investigator to numerically confirm the predictions being made, and will be released freely to the public as a much needed tool for studying L-functions. Many problems in number theory can be described in terms of the properties of so-called L-functions. These functions, which encode profound information about various number theoretic problems, have remained largely unyielding to mathematical analysis. Many deep problems in number theory would be solved if one could understand these functions in detail. Surprisingly, a seemingly unrelated field known as random matrix theory, a subject that originally arose in connection to experimental physics, has recently been found by number theorists and physicists alike to provide a framework in which to model the behavior of L-functions. This mysterious connection has been used successfully to make hitherto unimaginable predictions for the behavior of L-functions. The work in this proposal is concerned with exploring the connections between these two fields, number theory and random matrix theory. To assist in this project, the investigator is also preparing a software package, to be made freely available to the public, for numerically studying L-functions. This award is being cofunded by the Algebra, Number Theory, and Combinatorics Program, the Numeric, Symbolic, and Geometric Computation Program, and the Computational Mathematics Program.