Multiple Dirichlet Series and Number Theory

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This project concerns research in number theory. Classical problems in number theory include questions about the distribution of prime numbers, representations of a number as a sum of squares, and counting the number of lattice points in a circle or polygon in the plane. These questions have applications to such varied fields as cryptography, integer programming, complexity theory, and numerical integration. This research project will study several related questions through the unifying perspective of Dirichlet series in several complex variables. Integrated with the research activities will be continued mentorship of high school, undergraduate, and graduate students. The investigator will introduce these students to research-level mathematics through supervised research projects.This research project will investigate number-theoretic problems in four different areas, unified by the common theme of naturally occurring multiple Dirichlet series. The research in the first area, zeta functions of prehomogeneous vector spaces (PVS), is particularly timely due to the recent resurgence of interest in this field. A second topic is the study certain period integrals of Eisenstein series. Arithmetic consequences include explicit formulas for representation numbers of quadratic forms, which can in turn be reinterpreted as formulas for counting integral points on certain flag varieties. In a third direction, hyperbolic Fourier expansions of Eisenstein series motivate the construction of different type of multiple Dirichlet series which count totally positive elements in the ring of integers of a totally real number field. This can be reinterpreted as a lattice point counting problem in irrational polytopes. Fourthly, the project will study new types of subgroup growth and subring growth zeta functions. While this work should stimulate new research in subgroup growth, the initial motivation for study of these objects comes from complexity theory and lattice cryptography.