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arXiv · 2002.02640

A Pólya--Vinogradov inequality for short character sums

Matteo Bordignon
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Abstract

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

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Matteo Bordignon. 2021-02-19. A Pólya--Vinogradov inequality for short character sums. https://arxiv.org/abs/2002.02640

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