Khintchine's Constant

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

Let

 (1)

be the Simple Continued Fraction of a Real Number , where the numbers are Partial Quotients. Khintchine (1934) considered the limit of the Geometric Mean
 (2)

as . Amazingly enough, this limit is a constant independent of --except if belongs to a set of Measure 0-given by
 (3)

(Sloane's A002210), as proved in Kac (1959). The values are plotted above for to 500 and , , , the Euler-Mascheroni Constant , and the Copeland-Erdös Constant. Real Numbers for which include , , , and the Golden Ratio , plotted below.

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)

(Shanks and Wrench 1959). Gosper gave
 (8)

where is the Derivative of the Riemann Zeta Function. An extremely rapidly converging sum also due to Gosper is

 (9)
where is the Hurwitz Zeta Function.

Khintchine's constant is also given by the integral

 (10)

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

 (11)

for almost all Real (Lévy 1936, Finch). This number is sometimes called the Lévy Constant, and the argument of the exponential is sometimes called the Khintchine-Lévy Constant.

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

 (12)

Then
 (13)

for almost all real (Khintchine, Knuth 1981, Finch), and
 (14)

Furthermore, for , the limiting value
 (15)

exists and is a constant with probability 1 (Rockett and Szüsz 1992, Khintchine 1997).

See also Continued Fraction, Convergent, Khintchine-Lévy Constant, Lévy Constant, Partial Quotient, Simple Continued Fraction

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.