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

Matteo Bordignon

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

  \Big| \sum_{1 \le n\le N} \chi(n) \Big|\le c \sqrt{q} \log q, ∣ ∣ 1≤n≤N ∑ χ(n) ∣ ∣ ≤c q logq, where c=3/(4\pi^2)+o_q(1)c=3/(4π 2 )+o q (1) for even characters and c=3/(8\pi)+o_q(1)c=3/(8π)+o q (1) for odd characters, with explicit o_q(1)o q (1) terms. This improves a result of Frolenkov and Soundararajan for large qq. 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.