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

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:

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.