A1 Refereed original research article in a scientific journal
An explicit Pólya-Vinogradov inequality via Partial Gaussian sums
Authors: Matteo Bordignon, Bryce Kerr
Publisher: AMER MATHEMATICAL SOC
Publication year: 2020
Journal: Transactions of the American Mathematical Society
Journal name in source: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Journal acronym: T AM MATH SOC
Volume: 373
Issue: 9
First page : 6503
Last page: 6527
Number of pages: 25
ISSN: 0002-9947
eISSN: 1088-6850
DOI: https://doi.org/10.1090/tran/8138
Abstract
In this paper we obtain a new fully explicit constant for the Polya-Vinogradov inequality for squarefree modulus. Given a primitive character chi to squarefree modulus q, we prove the following upper bound:vertical bar Sigma(1 <= n <= N) chi(n)vertical bar <= c root 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.
In this paper we obtain a new fully explicit constant for the Polya-Vinogradov inequality for squarefree modulus. Given a primitive character chi to squarefree modulus q, we prove the following upper bound:vertical bar Sigma(1 <= n <= N) chi(n)vertical bar <= c root 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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