The maximum size of short character sums

In the present note, we prove new lower bounds on large values of character sums $\mathrm{\Delta }(x,q):={\max }_{\chi \mathrm{\ne }{\chi }_0}\mathrm{\mid }{\sum }_{n\le x}\chi (n)\mathrm{\mid }$ 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 $\log x = o(\sqrt{\log q})$\mathrm{.}Ourresultsareexpressedusingthecountingfunctionof$y$-friableintegerslessthan$x$whereweimprovethelevelofsmoothness$y$ for short intervals.