Title: Improving the Burgess bound via Polya-Vinogradov,
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.