On the Khintchine constant
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- by David H. Bailey, Jonathan M. Borwein and Richard E. Crandall PDF
- Math. Comp. 66 (1997), 417-431
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.References
- Roy Adler, Michael Keane, and Meir Smorodinsky, A construction of a normal number for the continued fraction transformation, J. Number Theory 13 (1981), no. 1, 95–105. MR 602450, DOI 10.1016/0022-314X(81)90031-7
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- David H. Bailey, Jonathan M. Borwein, and Roland Girgensohn, Experimental evaluation of Euler sums, Experiment. Math. 3 (1994), no. 1, 17–30. MR 1302815
- D. H. Bailey, “A Fortran-90 based multiprecision system,” ACM Trans. on Math. Software 21 (1995), 379–387.
- —, “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 mp-request@nas.nasa.gov, or by using Mosaic at address http://www.nas.nasa.gov.
- Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903, DOI 10.1007/978-1-4612-0965-2
- David Borwein, Jonathan M. Borwein, and Roland Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 277–294. MR 1335874, DOI 10.1017/S0013091500019088
- P. Borwein, “An efficient algorithm for the Riemann zeta function,” submitted for publication. Available from http://www.cecm.sfu/~pborwein.
- J. P. Buhler, R. E. Crandall, and R. W. Sompolski, Irregular primes to one million, Math. Comp. 59 (1992), no. 200, 717–722. MR 1134717, DOI 10.1090/S0025-5718-1992-1134717-4
- J. Buhler, R. Crandall, R. Ernvall, and T. Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp. 61 (1993), no. 203, 151–153. MR 1197511, DOI 10.1090/S0025-5718-1993-1197511-5
- R. Corless, personal communication.
- R. W. Gosper, personal communication.
- A. Khintchine, Continued fractions, University of Chicago Press, Chicago, 1964.
- D. Lehmer, “Note on an absolute constant of Khintchine,” Amer. Math. Monthly 46 (1939), 148-152.
- Leonard Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981. With a foreword by A. J. Van der Poorten. MR 618278
- Harald Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1172997, DOI 10.1137/1.9781611970081
- N. Nielsen, Die Gammafunktion, Princeton University Press, 1949.
- S. Plouffe, personal communication.
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Daniel Shanks and J. W. Wrench Jr., Khintchine’s constant, Amer. Math. Monthly 66 (1959), 276–279. MR 103167, DOI 10.2307/2309633
- P. Shiu, Computation of continued fractions without input values, Math. Comp. 64 (1995), no. 211, 1307–1317. MR 1297479, DOI 10.1090/S0025-5718-1995-1297479-9
- C.L. Siegel, Transcendental numbers, Chelsea, New York, 1965.
- Karl R. Stromberg, Introduction to classical real analysis, Wadsworth International Mathematics Series, Wadsworth International, Belmont, Calif., 1981. MR 604364
- Ilan Vardi, Computational recreations in Mathematica, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1991. MR 1150054
- T. Wieting, personal communication.
- John W. Wrench Jr., Further evaluation of Khintchine’s constant, Math. Comp. 14 (1960), 370–371. MR 170455, DOI 10.1090/S0025-5718-1960-0170455-1
- 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.
- D. Zagier, personal communication.
Additional Information
- David H. Bailey
- Affiliation: NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, California 94035-1000
- MR Author ID: 29355
- Email: dbailey@nas.nasa.gov
- Jonathan M. Borwein
- Affiliation: Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
- Email: jborwein@cecm.sfu.ca
- Richard E. Crandall
- Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202
- Email: crandall@reed.edu
- 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
- © Copyright Copyright Copyright
- Journal: Math. Comp. 66 (1997), 417-431
- MSC (1991): Primary 11Y60, 11Y65; Secondary 11M99
- DOI: https://doi.org/10.1090/S0025-5718-97-00800-4
- MathSciNet review: 1377659