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

• Matteo Bordignon ^[1]
  1. [1] University of New South Wales Canberra
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 38, Nº 4, 2022, págs. 1101-1127
• Idioma: inglés
• DOI: 10.4171/RMI/1328
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• Resumen

  • 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.