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Mathematics of Computation

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On the Khintchine constant

Authors: David H. Bailey, Jonathan M. Borwein and Richard E. Crandall
Journal: Math. Comp. 66 (1997), 417-431
MSC (1991): Primary 11Y60, 11Y65; Secondary 11M99
MathSciNet review: 1377659
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Abstract: We present rapidly converging series for the Khintchine constant and for general “Khintchine means” of continued fractions. We show that each of these constants can be cast in terms of an efficient free-parameter series, each series involving values of the Riemann zeta function, rationals, and logarithms of rationals. We provide an alternative, polylogarithm series for the Khintchine constant and indicate means to accelerate such series. We discuss properties of some explicit continued fractions, constructing specific fractions that have limiting geometric mean equal to the Khintchine constant. We report numerical evaluations of such special numbers and of various Khintchine means. In particular, we used an optimized series and a collection of fast algorithms to evaluate the Khintchine constant to more than 7000 decimal places.

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Additional Information

David H. Bailey
Affiliation: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, California 94035-1000
MR Author ID: 29355

Jonathan M. Borwein
Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada

Richard E. Crandall
Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202

Keywords: Khintchine constant, continued fractions, geometric mean, harmonic mean, computational number theory, zeta functions, polylogarithms
Received by editor(s): May 31, 1995
Received by editor(s) in revised form: February 8, 1996
Additional Notes: Research supported by the Shrum Endowment at Simon Fraser University and NSERC
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