Average first-passage times for character sums

Quanyu Tang, Hao Zhang

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_\varepsilon(p):=\min\{\ell\ge 1:\ S…