Sharp bounds for maximal sums of odd order Dirichlet characters

Alexander P. Mangerel

paper

Sharp bounds for maximal sums of odd order Dirichlet characters

arXiv:2505.07651

Abstract

Let $g \geq 3$ be fixed and odd, and for large $q$ let $I ¨ $ be a primitive Dirichlet character modulo $q$ of order $g$ . Conditionally on GRH we improve the existing upper bounds in the PÃ³lya-Vinogradov inequality for $I ¨ $ , showing that $M (I ¨ ) := t \geq 1 max n \leq t \sum I ¨ (n) ≪ q \frac{( lo g lo g q ) ^{1 - I^´_{g}} ( lo g lo g lo g lo g lo g q ) ^{I^´_{g}}}{( lo g lo g lo g q ) ^{1/4}},$ where $I^´_{g} := 1 - \frac{g}{I ¨} sin (I ¨ / g)$ . Furthermore, we show unconditionally that there is an infinite sequence of order $g$ primitive characters $I ¨ _{j}$ modulo $q_{j}$ for which $M (I ¨ _{j}) ≫ q_{j} \frac{( lo g lo g q _{j} ) ^{1 - I^´_{g}} ( lo g lo g lo g lo g lo g q _{j} ) ^{I^´_{g}}}{( lo g lo g lo g q _{j} ) ^{1/4}},$ 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.

14 pages, comments welcome; fixed typos