On the rapid computation of various polylogarithmic constants

Abstract

We give algorithms for the computation of the d-th digit of certain transcendental numbers in various bases. These algorithms can be easily implemented (multiple precision arithmetic is not needed), require virtually no memory, and feature run times that scale nearly linearly with the order of the digit desired. They make it feasible to compute, for example, the billionth binary digit of log (2) or π on a modest work station in a few hours run time.We demonstrate this technique by computing the ten billionth hexadecimal digit of π, the billionth hexadecimal digits of π 2 , log(2) and log 2 (2), and the ten billionth decimal digit of log(9/10).These calculations rest on the observation that very special types of identities exist for certain numbers like π, π 2 , log(2) and log 2 (2). These are essentially polylogarithmic ladders in an integer base. A number of these identities that we derive in this work appear to be new, for example the critical identity for π:.

Analysis of PSLQ, an integer relation finding algorithm

“…This algorithm in [2], [1] was called PSLQ and had the advantage of the adjustable parameter γ or τ . Applications and implementation of this earlier version of PSLQ(τ ) were described in [3], [7], [4]. These implementations showed that the parameters were a helpful feature of the algorithm.…”

“…One example is This remarkable series permits one to rapidly compute individual digits from the hexadecimal expansion of π. See [4] for details. It was found by applying PSLQ(τ ) to the vector X = (X 1 , X 2 , · · · , X 8 , π) where is in this lattice, so evidently X 7 is integrally dependent upon X 1 , .…”

“…However, in the wake of this numerical evidence, proofs have subsequently been found for many of these relations, including the above formula for π. See [3] and [4] for details.…”

A New Method for Symbolic Sequences Analysis. An Application to Long Sequences

“…In recent years new algorithms based on Bailey-Borwein-Plouffe (BBP) [14] have been discovered and highperformance computing tools have been developed. They made possible computation digits of very long sequences of several irrational numbers.…”

Ten Problems in Experimental Mathematics

“…This formula permits one to directly calculate binary or hexadecimal digits beginning at the n-th digit, without needing to calculate any of the first n − 1 digits [6]. This result has, in turn, led to more recent results that suggest a possible route to a proof that π and some other mathematical constants are 2-normal, or in other words that every m-long binary string occurs in the binary expansion with limiting frequency b −m [8,9].…”