Short Character Sums and the P\'{o}lya-Vinogradov Inequality

We show in a quantitative way that any odd character $\chi$ modulo $q$ of fixed order $g\ge 2$ satisfies the property that if the P'{o}lya-Vinogradov inequality for $\chi$ can be improved to $$\underset{1\le t\le q}{\max }\mid \sum _{n\le t}\chi (n)\mid =o_{q\to \mathrm{\infty }}(\sqrt{q}\log q)$$ then for any $ϵ>0$ one may exhibit cancellation in partial sums of $\chi$ on the interval $[1,t]$ whenever $t>q^{ϵ}$, i.e.,$$\sum _{n\le t}\chi (n)=o_{q\to \mathrm{\infty }}(t)\text{ for all }t>q^{ϵ}\text{.}$$ This generalizes and extends a result of Fromm and Goldmakher. We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing $g$ exhibit cancellation in short sums then the P'{o}lya-Vinogradov inequality can be improved for all odd primitive characters of order $g$. Some applications are also discussed.