| One of the central problems of analytic number theory is to bound the magnitude of sums of Dirichlet characters. The first breakthrough was made independently by Polya and Vinogradov in 1918; their result has never been improved, in general. However, under certain circumstances, improvements are known. For example, on the assumption of the Generalized Riemann Hypothesis (GRH), Montgomery and Vaughan improved the Polya-Vinogradov bound. Unconditionally, in a celebrated series of papers, Burgess improved the Polya-Vinogradov inequality for short character sums. However, long character sums remained out of reach until 2007, when Granville and Soundararajan showed that one can improve the Polya-Vinogradov theorem for characters of odd order. On the assumption of GRH, their methods can be adapted to improve the Montgomery-Vaughan bound, as well. This thesis builds on their work.;We first show (in Chapter II) how to refine the Granville-Soundararajan approach by introducing Halasz's results on mean values of multiplicative functions; this refinement leads to an improvement of their bound on odd order character sums. It is expected that the refinements presented here can be adapted to give a similar improvement on the assumption of GRH; this would conditionally prove a conjecture of Granville and Soundararajan, and be a best-possible result. However, this work is not yet complete.;In a different direction, we demonstrate (in Chapter III) how non-trivial bounds on very short character sums can be combined with the methods of Granville and Soundararajan to unconditionally improve the Polya-Vinogradov inequality. In particular, using results of Graham and Ringrose, and of Iwaniec-Gallagher-Postnikov, we improve the Polya-Vinogradov bound for characters of smooth or powerful conductor. |
