Sharp Bounds for Maximal Sums of Odd Order Dirichlet Characters

Abstract Let $g \geq 3$ be fixed and odd, and for large $q$ let $\chi $ be a primitive Dirichlet character modulo $q$ of order $g$. Conditionally on GRH we improve the existing upper bounds in the Pólya–Vinogradov inequality for $\chi $, showing that $$ \begin{align*} & M(\chi):= \max_{t \geq 1} \left|\sum_{n \leq t} \chi(n) \right| \ll \sqrt{q} \frac{(\log\log q)^{1-\delta_g} (\log\log\log\log\log q)^{\delta_g}}{(\log\log\log q)^{1/4}}, \end{align*} $$where $\delta _{g}:= 1-\tfrac{g}{\pi }\sin (\pi /g)$. Furthermore, we show unconditionally that there is an infinite sequence of order $g$ primitive characters $\chi _{j}$ modulo $q_{j}$ for which $$ \begin{align*} & M(\chi_j) \gg \sqrt{q_j} \frac{(\log\log q_j)^{1-\delta_g} (\log\log\log\log\log q_j)^{\delta_g}}{(\log\log\log q_j)^{1/4}}, \end{align*} $$so that our GRH bound is sharp up to the implicit constant. This improves on previous work of Granville and Soundararajan, of Goldmakher, and of Lamzouri and the author.