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arXiv · 2002.02640

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

Matteo Bordignon
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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.

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Matteo Bordignon. 2020-02-07. A P\'{o}lya--Vinogradov inequality for short character sums. https://arxiv.org/abs/2002.02640

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