Mathematical Sciences: Classical Analysis, Number Theory, and Supercomputers

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.