*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

Let

(1) |

(2) |

(3) |

The Continued Fraction for is [2, 1, 2, 5, 1, 1, 2, 1, 1, ...] (Sloane's A002211). It is not known if is
Irrational, let alone Transcendental. Bailey *et al. *(1995) have
computed to 7350 Digits.

Explicit expressions for include

(4) | |||

(5) | |||

(6) |

where is the Riemann Zeta Function and

(7) |

(8) |

(9) |

Khintchine's constant is also given by the integral

(10) |

If is the th Convergent of the Continued Fraction of , then

(11) |

Define the following quantity in terms of the th partial quotient ,

(12) |

(13) |

(14) |

(15) |

**References**

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. ``On the Khintchine Constant.'' *Math. Comput.* **66**, 417-431, 1997.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/khntchn/khntchn.html

Kac, M. *Statistical Independence and Probability, Analysts and Number Theory.* Providence, RI: Math. Assoc. Amer., 1959.

Khinchin, A. Ya. *Continued Fractions.* New York: Dover, 1997.

Knuth, D. E. Exercise 24 in *The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed.*
Reading, MA: Addison-Wesley, p. 604, 1981.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 46, 1983.

Lehmer, D. H. ``Note on an Absolute Constant of Khintchine.'' *Amer. Math. Monthly* **46**, 148-152, 1939.

Phillipp, W. ``Some Metrical Theorems in Number Theory.'' *Pacific J. Math.* **20**, 109-127, 1967.

Plouffe, S. ``Plouffe's Inverter: Table of Current Records for the Computation of Constants.'' http://www.lacim.uqam.ca/pi/records.html.

Rockett, A. M. and Szüsz, P. *Continued Fractions.* Singapore: World Scientific, 1992.

Shanks, D. and Wrench, J. W. ``Khintchine's Constant.'' *Amer. Math. Monthly* **66**, 148-152, 1959.

Sloane, N. J. A. Sequences
A002210/M1564
and A002211/M0118
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Vardi, I. ``Khinchin's Constant.'' §8.4 in *Computational Recreations in Mathematica.*
Reading, MA: Addison-Wesley, pp. 163-171, 1991.

Wrench, J. W. ``Further Evaluation of Khintchine's Constant.'' *Math. Comput.* **14**, 370-371, 1960.

© 1996-9

1999-05-26