Sharp bounds for maximal sums of odd order Dirichlet characters

Let $g\ge 3$ 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 $$M(\chi ):=\underset{t\ge 1}{\max }\mid \sum _{n\le t}\chi (n)\mid \ll \sqrt{q}\frac{(\log \log q{)}^{1-{\delta }_g}(\log \log \log \log \log q{)}^{{\delta }_g}}{(\log \log \log q{)}^{1\mathrm{/}4}},$$ where ${\delta }_g:=1-\frac{g}{\pi }\sin (\pi \mathrm{/}g)$. Furthermore, we show unconditionally that there is an infinite sequence of order $g$ primitive characters ${\chi }_j$ modulo $q_j$ for which $$M({\chi }_j)\gg \sqrt{q_j}\frac{(\log \log q_j{)}^{1-{\delta }_g}(\log \log \log \log \log q_j{)}^{{\delta }_g}}{(\log \log \log q_j{)}^{1\mathrm{/}4}},$$ 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.