An explicit Pólya-Vinogradov inequality via Partial Gaussian sums
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- by Matteo Bordignon and Bryce Kerr PDF
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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: \begin{align*} \left | \sum _{1 \leqslant n\leqslant N} \chi (n) \right |\leqslant c \sqrt {q} \log q, \end{align*} 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.References
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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
- 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.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 6503-6527
- MSC (2010): Primary 11L40, 11L05, 11H06, 11H60
- DOI: https://doi.org/10.1090/tran/8138
- MathSciNet review: 4155184