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

DOI:
https://doi.org/10.1090/S0025-5718-97-00800-4

MathSciNet review:
1377659

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**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`mp-request@@nas.nasa.gov`, or by using Mosaic at address`http://www.nas.nasa.gov`.**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.

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
11Y60,
11Y65,
11M99

Retrieve articles in all journals with MSC (1991): 11Y60, 11Y65, 11M99

Additional Information

**David H. Bailey**

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

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

DOI:
https://doi.org/10.1090/S0025-5718-97-00800-4

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