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

Abstract

We show in a quantitative way that any odd character $\chi$ modulo $q$ of fixed order $g \geq 2$ satisfies the property that if the P\'{o}lya-Vinogradov inequality for $\chi$ can be improved to $$\max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q)$$ 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.,$$\sum_{n \leq t} \chi(n) = o_{q \rightarrow \infty}(t) \text{ for all $t > q^{\epsilon}$.}$$ 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.