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 \begin{equation*} M(\chi )\ll _{\varepsilon } \sqrt {q}(\log q)^{1-\delta _g}(\log \log q)^{-1/4+\varepsilon }, \end{equation*} where $\delta _g ≔1-\frac {g}{\pi }\sin (\pi /g)$. This improves upon the works of Granville and Soundararajan [J. Amer. Math. Soc. 20 (2007), pp. 357–384] and of Goldmakher [Algebra Number Theory 6 (2012), pp. 123–163]. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that \begin{equation*} 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)}, \end{equation*} 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.