Constructions of L-functions, Eigenvalue Bounds and Statistics

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Daniel BumpStanford University UniversityDMS 9970841 This is a project in the study of automorphic forms, representation theory, and the statistics of unitary matrices. Professor Bump will study the symmetric cube and the symmetric fourth power L-functions by means of a new technique of double Dirichlet series. In this new approach L-functions are treated as residues of double Dirichlet series whose functional equations arise from functional equations of simpler L-functions. Other types of L-functions will be studied using the methods of Rankin-Selberg. In related work Professor Bump will also study the statistics of random unitary matrices via Shur functions. The investigator hopes that a novel new approach will further this classical study. This is a project in Number Theory. At its heart, Number Theory is the study of the counting numbers 1, 2, 3, 4.... These most basic numbers have elaborate, interesting, and mysterious properties with applications in many branches of science and modern technology. Many of the important properties of numbers can be mathematically summarized in objects called L-functions. These L-functions have their origins in the study of calculus and can be represented by several different formulas. Professor Bump's project concentrates on the study of certain kinds of L-functions. He expects his investigation of L-functions to increase our understanding of the properties of numbers that these functions encode.