Mathematical Sciences: Classical Analysis, Number Theory, and Supercomputers

这个项目的基本主题是试图理解在复杂分析和数论中出现的称为Zeta函数的函数类的根。有几组重要的Zeta函数需要分析：(I)欧拉积的线性组合；(Ii)Epstein Zeta函数(有理和无理类型)；(Iii)与余有限Fuchsian群相关的Selberg Zeta函数；(Iv)与基本区域或黎曼曲面上的Laplace算子的特征函数相关的L函数。将采用几种方法。一种涉及函数定义中参数的变化。当参数移动时，根跟踪从统计/动力学角度研究的路径。在函数形成有限维矢量空间的情况下，超级计算机分析也被用来解开复杂的关系。两个特定的目标是给出根的增长率，并确定是否所有的根都位于简单的子集上，例如直线。这项工作对数论中的基本和长期存在的问题有重要的应用。它还导致了超级计算在更多的数学分析理论领域的新用途，既作为一种实验工具，也可以在从函数定义中提取精确信息太难的情况下给出精确的概率估计。

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.