The convergence of certain Diophantine series
EntityUAM. Departamento de Matemáticas
10.1016/j.jnt.2021.04.022Journal of Number Theory 229 (2021): 179-198
Funded byThe 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)
ProjectGobierno de España. MTM2017-83496-P
SubjectsApproximate functional equation; Diophantine series; Lipschitz regularity; Matemáticas
Rights© 2021 Elsevier Inc.
Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.
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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