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arXiv · 2512.24631

Average first-passage times for character sums

Quanyu Tang·Hao Zhang
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Abstract

Let $\varepsilon>0$ and, for an odd prime $p$, set $$ S_\ell(p):=\sum_{n\le \ell}\left(\frac{n}{p}\right). $$ Define the first-passage time $$ f_\varepsilon(p):=\min\{\ell\ge 1:\ S_\ell(p)<\varepsilon\ell\}. $$ We prove that there exists a constant $c_\varepsilon>0$ such that, as $x\to\infty$, $$ \sum_{p\le x} f_\varepsilon(p)\sim c_\varepsilon \frac{x}{\log x}. $$

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Quanyu Tang, Hao Zhang. 2025-12-31. Average first-passage times for character sums. https://arxiv.org/abs/2512.24631

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