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On the convergence of Diophantine Dirichlet series

Part of: Zeta and $L$-functions: analytic theory Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  23 February 2012

T. Rivoal
T. Rivoal
Affiliation:
Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d'Hères cedex, France (tanguy.rivoal@fourier.ujf-grenoble.fr)
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Abstract

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We study various Dirichlet series of the form ∑n≥1f(πnα)/ns, where α is an irrational number and f(x) is a trigonometric function like cot(x), 1/sin(x) or 1/sin2(x). The convergence is slow and strongly depends on the Diophantine properties of α. We provide necessary and sufficient convergence conditions using the continued fraction of α. We also show that any one of our series is equal to a related series, which converges much faster, defined in term of iterations of the continued fraction operator α↦{1/α}.

Keywords

Diophantine approximationDirichlet seriescontinued fractions

MSC classification

Secondary: 11J70: Continued fractions and generalizations 11M41: Other Dirichlet series and zeta functions 11K70: Harmonic analysis and almost periodicity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 513 - 541
Copyright
Copyright © Edinburgh Mathematical Society 2012

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