The logarithm of irrational numbers and Beatty sequences

DOI10.4064/AA6695-1-2017zbMATH Open1429.11010arXiv1503.08512OpenAlexW2727950968MaRDI QIDQ5348020

Geremías Polanco

Publication date: 11 August 2017

Published in: Acta Arithmetica (Search for Journal in Brave)

Abstract: In this paper we find an identity that gives a representation for the logarithm of any two irrational numbers $a,b>1$ in terms of a series whose terms are ratios of elements from the Beatty Sequences generated by these two numbers. We also show that Sturmian sequences can be defined in terms of these ratios. Furthermore, we find an identity for such series that bears a superficial resemblance to (a discrete version of) Frullani's Integral.

Full work available at URL: https://arxiv.org/abs/1503.08512

zbMATH Keywords

logarithmSturmian sequencesBeatty sequencesseriesirrational numbersFrullani's integral

Mathematics Subject Classification ID

Convergence and divergence of series and sequences (40A05) Other combinatorial number theory (11B75) Arithmetic functions; related numbers; inversion formulas (11A25) Automata sequences (11B85)

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