JournalsrmiVol. 38, No. 4pp. 1101–1127

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

  • Matteo Bordignon

    University of New South Wales Canberra, Campbell, Australia
Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters cover
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Abstract

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

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

where c=3/(4π2)+oq(1)c=3/(4\pi^2)+o_q(1) for even characters and c=3/(8π)+oq(1)c=3/(8\pi)+o_q(1) for odd characters, with explicit oq(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.

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