The logarithm of irrational numbers and Beatty sequences
Volume 179 / 2017
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.
