# Improving the Burgess bound via Polya-Vinogradov

## TL;DR

This paper explores how slight enhancements in the Polya-Vinogradov inequality could lead to substantial improvements in Burgess' bound on character sums, using new bounds and relationships involving multiplicative functions.

## Contribution

It establishes a link between improvements in the Polya-Vinogradov inequality and potential advancements in Burgess' bound, utilizing recent bounds on character sums and properties of multiplicative functions.

## Key findings

- A lower bound on certain character sums derived from recent work.
- A quantitative relationship between mean and logarithmic mean of multiplicative functions.
- Implication that small improvements in Polya-Vinogradov could significantly enhance Burgess's bound.

## Abstract

We show that even mild improvements of the Polya-Vinogradov inequality would imply significant improvements of Burgess' bound on character sums. Our main ingredients are a lower bound on certain types of character sums (coming from works of the second author joint with J. Bober and Y. Lamzouri) and a quantitative relationship between the mean and the logarithmic mean of a completely multiplicative function.