Sharp bounds for maximal sums of odd order Dirichlet characters

TL;DR

This paper establishes sharp bounds for the maximum partial sums of odd order Dirichlet characters, improving existing inequalities under GRH and demonstrating the bounds' optimality through unconditional results.

Contribution

It provides new conditional upper bounds for maximal sums of odd order Dirichlet characters and proves their sharpness with unconditional lower bounds, advancing understanding of character sum behavior.

Findings

Conditional upper bounds under GRH for maximal sums.

Unconditional existence of characters reaching these bounds.

Improvement over previous bounds by Granville, Soundararajan, Goldmakher, and Lamzouri.

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 P\'{o}lya-Vinogradov inequality for $\chi$, showing that $$ M(\chi) := \max_{t \geq 1} \left|\sum_{n \leq t} \chi(n) \right| \ll \sqrt{q} \frac{(\log\log q)^{1-\delta_g} (\log\log\log\log\log q)^{\delta_g}}{(\log\log\log q)^{1/4}}, $$ 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 $$ M(\chi_j) \gg \sqrt{q_j} \frac{(\log\log q_j)^{1-\delta_g} (\log\log\log\log\log q_j)^{\delta_g}}{(\log\log\log q_j)^{1/4}}, $$ 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.