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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.07651 |
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Table of Contents:
- Let $g \geq 3$ be fixed and odd, and for large $q$ let $χ$ be a primitive Dirichlet character modulo $q$ of order $g$. Conditionally on GRH we improve the existing upper bounds in the Pólya-Vinogradov inequality for $χ$, showing that $$ M(χ) := \max_{t \geq 1} \left|\sum_{n \leq t} χ(n) \right| \ll \sqrt{q} \frac{(\log\log q)^{1-δ_g} (\log\log\log\log\log q)^{δ_g}}{(\log\log\log q)^{1/4}}, $$ where $δ_g := 1-\tfrac{g}π\sin(π/g)$. Furthermore, we show unconditionally that there is an infinite sequence of order $g$ primitive characters $χ_j$ modulo $q_j$ for which $$ M(χ_j) \gg \sqrt{q_j} \frac{(\log\log q_j)^{1-δ_g} (\log\log\log\log\log q_j)^{δ_g}}{(\log\log\log q_j)^{1/4}}, $$ 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.