## 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

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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
- 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