Average first-passage times for character sums

Abstract

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

9 pages. v2: added a reference and corrected several typos