The maximum size of short character sums

In the present note, we prove new lower bounds on large values of character sums Δ(x,q):=maxχ≠χ0|∑n≤xχ(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 logx=o(logq). 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.