Long large character sums

Abstract

In this paper, we prove a lower bound for $\underset{\chi \ne {\chi }_0}{\max }|{\sum }_{n\le x}\chi (n)|$, when $x=\frac{q}{(\log q{)}^B}$. This improves on a result of Granville and Soundararajan for large character sums when the range of summation is wide. When $B$ goes to zero, our lower bound recovers the expected maximal value of character sums for most characters.

27 pages