Short Character Sums and the Pólya-Vinogradov Inequality

Abstract

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 Pólya-Vinogradov inequality for $\chi$ can be improved to $$\underset{1\le t\le q}{\max }\left|\sum _{n\le t}\chi (n)\right|=o_{q\to \infty }(\sqrt{q}\log q)$$ then for any $\epsilon >0$ one may exhibit cancellation in partial sums of $\chi$ on the interval $[1,t]$ whenever $t>q^{\epsilon }$, i.e.,$$\sum _{n\le t}\chi (n)=o_{q\to \infty }(t)\text{ for all t>qε.}$$ 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 Pólya-Vinogradov inequality can be improved for all odd primitive characters of order $g$. Some applications are also discussed.

25 pages