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Mathematics of Computation

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On the rapid computation of various polylogarithmic constants

Authors: David Bailey, Peter Borwein and Simon Plouffe
Journal: Math. Comp. 66 (1997), 903-913
MSC (1991): Primary 11A05, 11Y16, 68Q25
MathSciNet review: 1415794
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Abstract | References | Similar Articles | Additional Information

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 $\pi $ on a modest work station in a few hours run time. We demonstrate this technique by computing the ten billionth hexadecimal digit of $\pi $, the billionth hexadecimal digits of $\pi ^{2}, \; \log (2)$ 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 $\pi $, $\pi ^{2}$, $\log (2)$ 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 $\pi $:

\begin{equation*}\pi = \sum _{i=0}^{\infty }\frac {1}{16^{i}}\bigr ( \frac {4}{8i+1} - \frac {2}{8i+4} - \frac {1}{8i+5} - \frac {1}{8i+6} \bigl ).\end{equation*}

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Additional Information

David Bailey
Affiliation: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, California 94035-1000

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6

Simon Plouffe
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6

Keywords: Computation, digits, log, polylogarithms, SC, $\pi $, algorithm
Received by editor(s): October 11, 1995
Received by editor(s) in revised form: February 16, 1996
Additional Notes: Research of the second author was supported in part by NSERC of Canada.
Article copyright: © Copyright 1997 American Mathematical Society