Long large character sums

Abstract

In this paper, we prove a lower bound for max χ ≠ χ 0 | ∑ n ⩽ x χ ( n ) | ${\max_{\chi \ne \chi _0}} |\sum _{n\leqslant x} \chi (n) |$ , when x = q / ( log q ) B $x= {q}/{(\log q)^B}$ . 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.