On certain exponential and character sums

Abstract

This thesis considers four distinct problems in the area of exponential and character sums. The strategy used to approach each problem can be considered as falling roughly into what is known as Vinogradov's method. The problems considered are rational exponential sums over the divisor function, character sums over shifted primes, character sums mixed with the exponential of a polynomial and the fourth moment of character sums. For each problem we obtain new quantitative bounds for the relevant sums which either improve or extend existing results except for the case of rational exponential sums over the divisor function whose solution has not been considered before although has been posed by Shparlinski.