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The convergence of certain Diophantine series

Author
Chamizo Lorente, Fernandountranslated; Martin, Bruno
Entity
UAM. Departamento de Matemáticas
Publisher
Elsevier
Date
2021-06-01
Citation
Journal of Number Theory 229 (2021): 179-198
ISSN
0022-314X (print)
DOI
10.1016/j.jnt.2021.04.022
Funded by
The first author is partially supported by the MTM2017-83496-P grant of the MICINN (Spain) and by “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554). The second author is supported by ArithRand, a joint Austrian-French bilateral project (Austrian Science Fund FWF: I 4945-N, Agence Nationale de la Recherche ANR: ANR-20-CE91-0006)
Project
Gobierno de España. MTM2017-83496-P
Editor's Version
https://doi.org/10.1016/j.jnt.2021.04.022
Subjects
Approximate functional equation; Diophantine series; Lipschitz regularity; Matemáticas
URI
http://hdl.handle.net/10486/700618
Rights
© 2021 Elsevier Inc.

Licencia de Creative Commons
Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Abstract

For x irrational, we study the convergence of series of the form ∑n−sf(nx) where f is a real-valued, 1-periodic function which is continuous, except for singularities at the integers with a potential growth. We show that it is possible to fully characterize the convergence set and to approximate the series in terms of the continued fraction of x. This improves and generalizes recent results by Rivoal who studied the examples f(t) = cot⁡(πt) and f(t) = sin−2⁡(πt)
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Google™ Scholar:Chamizo Lorente, Fernando - Martin, Bruno

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