Large odd order character sums and improvements of the Pólya-Vinogradov inequality

Abstract

For a primitive Dirichlet character $\chi$ modulo $q$, we define $M(\chi )={\max }_t|{\sum }_{n\le t}\chi (n)|$. In this paper, we study this quantity for characters of a fixed odd order $g\ge 3$. 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 $$M(\chi ){\ll }_{\epsilon }\sqrt{q}(\log q{)}^{1-{\delta }_g}(\log \log q{)}^{-1/4+\epsilon },$$ 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.

34 pages, fixed some typos