A Pólya–Vinogradov inequality for short character sums

• Published in: Canadian mathematical bulletin Vol. 64; no. 4; pp. 906 - 910
• Main Author: Bordignon, Matteo
• Format: Journal Article
• Language: English
• Published: Canada Canadian Mathematical Society 01.12.2021; Cambridge University Press
• Subjects: Inequality; 11L05; Character sums; 11L40; Pólya-Vinogradov inequality
• ISSN: 0008-4395; 1496-4287
• DOI: 10.4153/S0008439520000934

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.