# Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters

### Matteo Bordignon

University of New South Wales Canberra, Campbell, Australia

## Abstract

In this paper we obtain a new fully explicit constant for the Pólya–Vinogradov inequality for primitive characters. Given a primitive character $\chi$ modulo $q$, we prove the following upper bound:

$\Big| \sum_{1 \le n\le N} \chi(n) \Big|\le c \sqrt{q} \log q,$

where $c=3/(4\pi^2)+o_q(1)$ for even characters and $c=3/(8\pi)+o_q(1)$ for odd characters, with explicit $o_q(1)$ terms. This improves a result of Frolenkov and Soundararajan for large $q$. We proceed, following Hildebrand, to obtain the explicit version of a result by Montgomery–Vaughan on partial Gaussian sums and an explicit Burgess-like result on convoluted Dirichlet characters.

## Cite this article

Matteo Bordignon, Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters. Rev. Mat. Iberoam. 38 (2022), no. 4, pp. 1101–1127

DOI 10.4171/RMI/1328