An explicit Pólya-Vinogradov inequality via Partial Gaussian sums

Bordignon, Matteo; Kerr, Bryce

doi:10.1090/tran/8138

An explicit Pólya-Vinogradov inequality via Partial Gaussian sums

Authors: Matteo Bordignon and Bryce Kerr
Journal: Trans. Amer. Math. Soc. 373 (2020), 6503-6527
MSC (2010): Primary 11L40, 11L05, 11H06, 11H60
DOI: https://doi.org/10.1090/tran/8138
Published electronically: July 8, 2020
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Abstract: In this paper we obtain a new fully explicit constant for the Pólya-Vinogradov inequality for squarefree modulus. Given a primitive character $\chi$ to squarefree modulus , we prove the following upper bound:

$\displaystyle \left \vert \sum _{1 \le n\le N} \chi (n) \right \vert\le c \sqrt {q} \log q,$

where $c=1/(2\pi ^2)+o(1)$ for even characters and $c=1/(4\pi )+o(1)$ for odd characters, with an explicit

term. This improves a result of Frolenkov and Soundararajan for large

. We proceed via partial Gaussian sums rather than the usual Montgomery and Vaughan approach of exponential sums with multiplicative coefficients. This allows a power saving on the minor arcs rather than a factor of $\log {q}$ as in previous approaches and is an important factor for fully explicit bounds.

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