A class of digit extraction BBP-type formulas in general binary bases

Abstract

BBP-type formulas are usually discovered experimentally, one at a time and in specific bases, through computer searches. In this paper, however, we derive directly, without doing any searches, explicit digit extraction BBP-type formulas in general binary bases $b=2^{12p}$, for $p$ positive odd integers. As particular examples, new binary formulas are presented for $\pi \sqrt{3}$, $\pi \sqrt{3}\log 2$, $\sqrt{3}{\mathrm{Cl}}_2(\pi /3)$ and a couple of other polylogarithm constants. A variant of the formula for $\pi \sqrt{3}\log 2$ derived in this paper has been known for over ten years but was hitherto unproved. Binary BBP-type formulas for the logarithms of an infinite set of primes and binary BBP-type representations for the arctangents of an infinite set of rational numbers are also presented. Finally, new binary BBP-type zero relations are established.

arXiv admin note: text overlap with arXiv:1603.05810