Hilbert's Inequality, Generalized Factorials, and Partial Factorizations of Generalized Binomial Products

Abstract

This dissertation treats three topics in number theory. The first topic concerns the problem of determining the optimal constant in the Montgomery–Vaughan weighted generalization of Hilbert's inequality. The second topic presents a further generalization of Bhargava's generalized factorials in the ring Z. We define invariants associated to all pairs (S,b) of a nonempty subset S of Z and a nontrivial proper ideal b in Z and use them to construct generalized factorials. The third topic is asymptotics of partial factorizations of products of generalized binomial coefficients constructed using generalized factorials from the second topic.