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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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  • 1. R. Adler, M. Keane and M. Smorodinsky, ``A construction of a normal number for the continued fraction transformation,'' J. Number Theory 13 (1981), 95-105. MR 82k:10070
  • 2. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, Dover Publications, Inc., New York, 1972. MR 94b:00012
  • 3. D. H. Bailey, J. M. Borwein, and R. Girgensohn, ``Experimental evaluation of Euler sums,'' Experimental Math. 3 (1994), 17-30. MR 96e:11168
  • 4. D. H. Bailey, ``A Fortran-90 based multiprecision system,'' ACM Trans. on Math. Software 21 (1995), 379-387.
  • 5. -, ``Multiprecision translation and execution of Fortran programs,'' ACM Trans. on Math. Software 19 (1993), 288-319. This software and documentation, as well as that described in [4], may be obtained by sending electronic mail to, or by using Mosaic at address
  • 6. B.C. Berndt, Ramanujan's notebooks, Part III, Springer Verlag, New York, 1991. MR 92j:01069
  • 7. D. Borwein, J.M. Borwein, and R. Girgensohn, ``Explicit evaluation of Euler sums,'' Proc. Edinburgh Math. Soc. 38 (1995), 277-294. MR 96f:11106
  • 8. P. Borwein, ``An efficient algorithm for the Riemann zeta function,'' submitted for publication. Available from http://www.cecm.sfu/~pborwein.
  • 9. J. Buhler, R. Crandall, and R. Sompolski, ``Irregular primes to one million,'' Math. Comp. 59 (1992), 717-722. MR 93a:11106
  • 10. J. Buhler, R. Crandall, R. Ernvall, and T. Metsänkylä, ``Irregular primes and cyclotomic invariants to four million,'' Math. Comp. 61 (1993), 151-153. MR 93k:11014
  • 11. R. Corless, personal communication.
  • 12. R. W. Gosper, personal communication.
  • 13. A. Khintchine, Continued fractions, University of Chicago Press, Chicago, 1964.
  • 14. D. Lehmer, ``Note on an absolute constant of Khintchine,'' Amer. Math. Monthly 46 (1939), 148-152.
  • 15. L. Lewin, Polylogarithms and associated functions, North Holland, New York, 1981. MR 83b:33019
  • 16. H. Niederreiter, Random number generation and quasi-Monte-Carlo methods, CBMS-NSF Regional Conf. Ser. Appl. Math., vol. 63, 1992. MR 93h:65008
  • 17. N. Nielsen, Die Gammafunktion, Princeton University Press, 1949.
  • 18. S. Plouffe, personal communication.
  • 19. C. Ryll-Nardzewski, ``On the ergodic theorems (I,II),'' Studia Math. 12 (1951) 65-79. MR 13:757a; MR 13:757b
  • 20. D. Shanks and J. W. Wrench, ``Khintchine's constant,'' Amer. Math. Monthly 66 (1959), 276-279. MR 21:1950
  • 21. P. Shiu, ``Computation of continued fractions without input values,'' Math. Comp. 64 (1995), 1307-1317. MR 96b:11165
  • 22. C.L. Siegel, Transcendental numbers, Chelsea, New York, 1965.
  • 23. K. R. Stromberg, An introduction to classical real analysis, Wadsworth, Belmont, CA, 1981. MR 82c:26002
  • 24. I. Vardi, Computational recreations in mathematica, Addison-Wesley, Redwood City, CA, 1991. MR 93e:00002
  • 25. T. Wieting, personal communication.
  • 26. J. W. Wrench, ``Further evaluation of Khintchine's constant,'' Math. Comp. 14 (1960), 370-371. MR 30:693
  • 27. J. W. Wrench and D. Shanks, ``Questions concerning Khintchine's constant and the efficient computation of regular continued fractions,'' Math. Comp. 20 (1966), 444-448.
  • 28. D. Zagier, personal communication.

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Additional Information

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

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
Article copyright: © Copyright Copyright Copyright

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