Integer Ratios of Factorials, Hypergeometric Functions, and Related Step Functions.

Abstract

In this thesis we study the question of certain sequences of ratios products of factorials are always integers. Equivalently, this is a study of when certain step functions related to the Beurling--Nyman criterion for the Riemann Hypothesis are always nonnegative.

We give a complete classification of what we call the "height 1" case, which proves a conjecture of V. I. Vasyunin regarding certain step functions that only take the values 0 and 1. For larger height we give partial results: we prove a conjecture of A. Borisov that, for fixed height, the range of values taken by one of the step functions we consider increases with its length; additionally, we prove that for larger heights there exists a classification similar to that for height 1. In addition to the application of these theorems to the classification of integer factorial ratios and nonnegative step functions related to the Beurling--Nyman criterion, through work of A. Borisov, these results have applications to the classification of cyclic quotient singularities.

These results are proved using a variety of methods. These include Beukers and Heckman's classification of algebraic hypergeometric functions, complex analysis techniques familiar to analytic number theory, and a theorem of Jim Lawrence about closed subgroups of the torus.