Short Character Sums and Their Applications

In analytic number theory, the most natural generalisations of the famous Riemann zeta function are the Dirichlet L-functions. Each Dirichlet L-function is attached to a q-periodic arithmetic function for some natural number q, known as a Dirichlet character modulo q. Dirichlet characters and L-functions encode with them information about the primes, especially in reference to their remainders modulo q. For this reason, number theorists are often interested in bounds on short sums of a Dirichlet character over the integers. For one, such sums appear as an intermediate step in partial summation bounds for Dirichlet L-functions. However, short character sum estimates may also be used directly to tackle other number theoretic problems. In this thesis, we wish to examine the application of character sum estimates in some specific settings. There are three main estimates that we are interested in: the trivial bound, Burgess’ bound, and the Pólya–Vinogradov inequality. Of these, we will focus primarily on Burgess’ bound; the main result of this thesis will be the computation of explicit constants versions of Burgess’ bound for a variety of parameters, improving upon the work of Treviño [54]. The interplay between Burgess’ bound and the Pólya–Vinogradov inequality is vital in this explicit setting, and we will dedicate a portion of this thesis to investigating these interactions. This makes precise the work of Fromm and Goldmakher [21], who demonstrated a counterintuitive influence the Pólya–Vinogradov inequality has on Burgess’ bound. Once we have established improvements to Burgess’ bound in the explicit setting, we will “test” these improvements by tackling several applications where Burgess’ bound has been used previously. Primary among these is an explicit bound for L-functions across the critical strip. We also include applications to norm-Euclidean cyclic fields and least kth power non-residues.