Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.7

Generalized Continued Logarithms and Related Continued Fractions

Jonathan M. Borwein
University of Newcastle
Building V
University Drive
Callaghan NSW 2308

Kevin G. Hare and Jason G. Lynch
Department of Pure Mathematics
University of Waterloo
200 University Ave. W.
Waterloo, ON N2L 3G1


We study continued logarithms, as introduced by Gosper and studied by Borwein et al. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et al., we focus on a new generalization to an arbitrary integer base b. We show that all of our so-called type III continued logarithms converge and all rational numbers have finite type III continued logarithms. As with simple continued fractions, we show that the continued logarithm terms, for almost every real number, follow a specific distribution. We also generalize Khinchin's constant from simple continued fractions to continued logarithms, and show that these logarithmic Khinchin constants have an elementary closed form. Finally, we show that simple continued fractions are the limiting case of our continued logarithms, and briefly consider how we could generalize beyond continued logarithms.

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Received October 3 2016; revised versions received May 8 2017; May 9 2017. Published in Journal of Integer Sequences, May 21 2017.

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