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

Abstract: In this paper we obtain a new fully explicit constant for the P\'olya-Vinogradov inequality for primitive characters. Given a primitive character $\chi$ modulo $q$, we prove the following upper bound \begin{align*} \left| \sum_{1 \le n\le N} \chi(n) \right|\le c \sqrt{q} \log q, \end{align*} 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, obtaining the explicit version of a result by Montgomery--Vaughan on partial Gaussian sums and an explicit Burgess-like result on convoluted Dirichlet characters.