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The logarithm of irrational numbers and Beatty sequences

Volume 179 / 2017

Geremias Polanco Acta Arithmetica 179 (2017), 101-123 MSC: Primary 11A25, 11B75, 40A05; Secondary 11B85. DOI: 10.4064/aa6695-1-2017 Published online: 7 July 2017

Abstract

We find an identity that expresses the logarithm of the ratio of any two irrational numbers $a, b \gt 1$ via a series whose terms are ratios of elements of the Beatty sequences generated by $a$ and $b$. 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.

Authors

  • Geremias PolancoSchool of Natural Science
    Hampshire College
    893 West St.
    Amherst, MA 01002, U.S.A.
    and
    Facultad de Ciencias
    Instituto de Matemáticas
    Universidad Autónoma de Santo Domingo
    Av. Alma Mater
    Santo Domingo 10105, Dominican Republic
    e-mail

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