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On the Khintchine Constant

Bailey, David H. and Borwein, Jonathan M. and Crandall, Richard E. (1997) On the Khintchine Constant. Mathematics of Computation, 66 (217). pp. 417-431.

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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 of which involves only 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.

      Item Type: Article
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: Khintchine constant, continued fractions, geometric mean, harmonic mean, computational number theory, zeta functions, polylogarithms
      Subjects: 11-xx Number theory > 11Mxx Zeta and $L$-functions: analytic theory
      11-xx Number theory > 11Yxx Computational number theory
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 20 Nov 2003
      Last Modified: 13 Sep 2014 20:51
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/102

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