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

TL;DR

This paper derives explicit BBP-type digit extraction formulas in general binary bases without search, providing new formulas for constants like pi times sqrt(3) and log of primes, along with related zero relations.

Contribution

It introduces a direct derivation method for BBP-type formulas in binary bases, expanding the set of known formulas for various mathematical constants.

Findings

Explicit formulas for pi*sqrt(3), pi*sqrt(3)*log(2), and related constants.

Binary BBP formulas for logs of primes and arctangents of rationals.

New zero relations for BBP-type formulas.

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 $\pi\sqrt 3$, $\pi\sqrt 3\log 2$, $\sqrt 3\;{\rm Cl}_2(\pi/3)$ 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.