Title: Large odd order character sums and improvements of the Pólya-Vinogradov inequality,
Abstract: For a primitive Dirichlet character $\chi$ modulo $q$, we define
$M(\chi)=\max_{t } |\sum_{n \leq t} \chi(n)|$. In this paper, we study this
quantity for characters of a fixed odd order $g\geq 3$. Our main result
provides a further improvement of the classical Pólya-Vinogradov inequality
in this case. More specifically, we show that for any such character $\chi$ we
have $$M(\chi)\ll_{\varepsilon} \sqrt{q}(\log q)^{1-\delta_g}(\log\log
q)^{-1/4+\varepsilon},$$ where $\delta_g := 1-\frac{g}{\pi}\sin(\pi/g)$. This
improves upon the works of Granville and Soundararajan and of Goldmakher.
Furthermore, assuming the Generalized Riemann hypothesis (GRH) we prove that $$
M(\chi) \ll \sqrt{q} \left(\log_2 q\right)^{1-\delta_g} \left(\log_3
q\right)^{-\frac{1}{4}}\left(\log_4 q\right)^{O(1)}, $$ where $\log_j$ is the
$j$-th iterated logarithm. We also show unconditionally that this bound is best
possible (up to a power of $\log_4 q$). One of the key ingredients in the proof
of the upper bounds is a new Halász-type inequality for logarithmic mean
values of completely multiplicative functions, which might be of independent
interest.