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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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 $ q$, 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 $ o(1)$ term. This improves a result of Frolenkov and Soundararajan for large $ q$. 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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Additional Information

Matteo Bordignon
Affiliation: School of Science, The University of New South Wales Canberra, Australia
MR Author ID: 1324990
Email: m.bordignon@student.adfa.edu.au

Bryce Kerr
Affiliation: Department of Mathematics and Statistics, University of Turku, Turku, Finland
Email: bryce.kerr@utu.fi

DOI: https://doi.org/10.1090/tran/8138
Received by editor(s): September 3, 2019
Received by editor(s) in revised form: January 22, 2020
Published electronically: July 8, 2020
Additional Notes: The second author was supported by Australian Research Council Discovery Project DP160100932.
Article copyright: © Copyright 2020 American Mathematical Society