In this paper we obtain a new fully explicit constant for the Pólya–Vinogradov inequality for primitive characters. Given a primitive character modulo , we prove the following upper bound:
where for even characters and for odd characters, with explicit terms. This improves a result of Frolenkov and Soundararajan for large . 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.
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Matteo Bordignon, Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters. Rev. Mat. Iberoam. 38 (2022), no. 4, pp. 1101–1127DOI 10.4171/RMI/1328