A P\'{o}lya--Vinogradov inequality for short character sums

Abstract

In this paper we obtain a variation of the P\'{o}lya--Vinogradov inequality with the sum restricted to a certain height. Assume $\chi$ to be a primitive character modulo $q$, $\epsilon > 0$ and $N\le q^{1-\gamma}$, with $0\le \gamma \le 1/3$. We prove that \begin{equation*} \left|\sum_{n=1}^N \chi(n) \right|\le c(\frac{1}{3}-\gamma+\epsilon)\sqrt{q}\log q \end{equation*} with $c=2/\pi^2+o(1)$ if $\chi$ is even and $c=1/\pi+o(1)$ if $\chi$ is odd.