Mathematical Sciences: Classical Analysis, Number Theory, and Supercomputers

这个项目的基本主题是试图 理解函数类的根源， 复分析和数论中的zeta函数 那里 是要分析的几组重要的zeta函数：（i） 线性组合的欧拉产品;（ii）爱泼斯坦泽塔 （三）合理和不合理的功能; 与上有限Fuchsian群相关的Selberg zeta函数 (iv)与拉普拉斯本征函数相关的L-函数 操作员接管基本域或黎曼曲面。 将采用几种方法。 一个涉及的变化， 函数定义中的参数。 作为参数 移动，根跟踪路径，从一个研究 统计/动力学观点。 超级计算机分析是 也被用来解开复杂的关系， 函数形成有限维向量空间。 两个特定 目标是给出根的生长率， 是否所有的根都位于简单的子集上，例如直线。 这项工作具有重要的应用基础和长期- 数论中的常见问题 这也导致了新的用途 超级计算在数学理论领域的应用 分析，既作为一种实验工具，也可以给 精确概率估计 信息太难从函数中提取， 定义.

This underlying theme of this project is an attempt to understand the roots of classes of functions which arise in complex analysis and number theory called zeta functions. There are several important sets of zeta functions to be analyzed: (i) linear combinations of Euler products; (ii) Epstein zeta functions (of both the rational and irrational type); (iii) Selberg zeta functions associated with cofinite Fuchsian groups; (iv) L- functions associated with eigenfunctions of the Laplace operator taken over fundamental domains or Riemann surfaces. Several methods will be employed. One involves the variation of parameters in the definition of functions. As the parameters move about, the roots trace paths which are studied from a statistical/dynamical point of view. Supercomputer analysis is also used to unravel complex relationships in cases where functions form finite dimensional vector spaces. Two particular goals are to give growth rates of the roots and to determine whether or not all roots lie on simple subsets such as lines. This work has important application to fundamental and long- standing questions in number theory. It also leads to new uses of supercomputing in the more theoretical areas of mathematical analysis, both as an experimental tools and one which can give sharp probability estimates in situations where precise information is too difficult to extract from the functional definitions.