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A Pólya–Vinogradov inequality for short character sums
Canadian Mathematical Bulletin ( IF 0.5 ) Pub Date : 2020-12-02 , DOI: 10.4153/s0008439520000934
Matteo Bordignon

In this paper, we obtain a variation of the Pólya–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 $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$ with $c=2/\pi ^2$ if $\chi $ is even and $c=1/\pi $ if $\chi $ is odd. The result is based on the work of Hildebrand and Kerr.



中文翻译:

短字符总和的 Pólya-Vinogradov 不等式

在本文中,我们获得了 Pólya-Vinogradov 不等式的变体,其总和限制在某个高度。假设 $\chi $ 是一个原始字符模q $ \epsilon>0$ $N\le q^{1-\gamma }$ $0\le \gamma \le 1/3$ 。我们证明 $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q} \log q \end{align*} $$ $c=2/\pi ^2$ 如果 $\chi $ 是偶数并且 $c=1/\pi $ 如果 $\chi $ 是奇数。结果基于 Hildebrand 和 Kerr 的工作。

更新日期:2020-12-02
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