arXiv:2611.08816v1 Announce Type: new Abstract: Let $S_{\chi}(N)= \sum_{n\leq N} \chi(n)$, where $\chi$ is a Dirichlet character modulo a prime $p$. For integral $r\geq 2$, Burgess gave bounds of the shape $S_{\chi}(N) \ll N^{1-1/r} p^{(r+1)/(4r^{2})} \log p$. The factor of $\log p$ was reduced by several authors. We show that it can be removed completely.
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