|Date: Monday, November 20, 2017
Location: 4088 East Hall (4:10 PM to 5:30 PM)
Title: The unreasonable effectiveness of the Polya-Vinogradov inequality
Abstract: The Polya-Vinogradov inequality, an upper bound on character sums proved a century ago, is essentially best-possible. Unfortunately, it's also not so useful in applications, since it's nontrivial only on long sums (while in practice one usually needs estimates on sums which are as short as possible). The best tool we have to handle shorter sums is the Burgess bound, discovered in 1957; this is generally considered to supersede Polya-Vinogradov, both because its proof is "deeper" (building on results from algebraic geometry) and because it is more applicable.
In this talk I will introduce and motivate both of these bounds, and then describe the unexpected result (joint with Elijah Fromm, Williams '17) that even a tiny improvement of the (allegedly weaker) Polya-Vinogradov inequality would imply a major improvement of the (supposedly superior) Burgess bound. I'll also talk about a related connection between improving Polya-Vinogradov and the classical problem of bounding the least quadratic nonresidue (joint with Jonathan Bober, University of Bristol).
Speaker: Leo Goldmakher
Institution: Williams College
Event Organizer: Wei Ho