Explicit bounds on Exponential Sums of Hecke Eigenvalues, on the modular group SL(2, Z) and congruence subgroups.

Abstract

Bounds on exponential sums have been an important topic of study in Analytic Number Theory, offering solutions to many problems, including the famous Ternary Goldbach Conjecture and Fermat's Last Theorem. In 2006, L. Zhao published a paper finding bounds on a particular exponential sum with square root amplitude, twisted with Hecke-eigenvalues over the full modular group. In this talk I will present some history of these sums, as well as demonstrating more explicit constants for the bounds found by Zhao, and finally an extension of these bounds to the congruence subgroups.