The maximum size of short character sums

Abstract

In the present note, we prove new lower bounds on large values of character sums $\Delta (x,q):={\max }_{\chi \ne {\chi }_0}|{\sum }_{n\le x}\chi (n)|$ in certain ranges of $x$. Employing an implementation of the resonance method developed in a work involving the author in order to exhibit large values of $L$- functions, we improve some results of Hough in the range $\log x=o(\sqrt{\log q})$. Our results are expressed using the counting function of $y$- friable integers less than $x$ where we improve the level of smoothness $y$ for short intervals.