Khinchin's Constant

Let

(1)

be the simple continued fraction of a "generic" real number , where the numbers  are the partial quotients. Khinchin (1934) considered the limit of the geometric mean

(2)

as . Amazingly, except for a set of measure 0, this limit is a constant independent of  given by

(3)

(OEIS A002210), as proved in Kac (1959).

The constant is known as Khinchin's constant, and is commonly also spelled "Khintchine's constant" (Shanks and Wrench 1959, Bailey et al. 1997).

It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of  is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.

It is not known if  is irrational, let alone transcendental.

While it is known that almost all numbers  have limits  that approach , this fact has not been proven for any explicit real number , e.g., a real number cast in terms of fundamental constants (Bailey et al. 1997).

The values  are plotted above for  to 500 and , , the Euler-Mascheroni constant , and the Copeland-Erdős constant . Interestingly, the shape of the curves is almost identical to the corresponding curves for the Lévy constant.

If  is the th convergent of the continued fraction of , then

			(4)
			(5)
			(6)

(OEIS A086702) for almost all real  (Lévy 1936, Finch 2003), where  is the natural logarithm of 2. This number is sometimes called the Lévy constant.

Product expressions for  include

(7)

(Shanks and Wrench 1959; Khinchin 1997, p. 93; Borwein and Bailey 2003, p. 25; Havil 2003, p. 161), where  is the natural logarithm, and

(8)

where  is the third-order finite difference operator, the latter of which obtained from three applications of summation by parts to the (logarithm of the) usual product definition (W. Gosper, pers. comm. Nov. 14, 2017).

Products such as these can be converted to sums by taking the logarithm of both sides and using . Sums for  include

(9)

where  is the Riemann zeta function and  is an alternating harmonic number (Bailey et al. 1997),

(10)

where  is the derivative of the Riemann zeta function (Gosper, pers. comm., Jun. 25, 1996) and the extremely rapidly converging sum originally due to Gosper (pers. comm., Jun. 25, 1996) and streamlined by O. Pavlyk (pers. comm., Apr. 24, 2006) is given by

(11)

where

(12)

 is a Hurwitz zeta function, , and  is a hypergeometric function.

Khinchin's constant is also given by the integrals

			(13)
			(14)

(Shanks and Wrench 1959) and

(15)

Corless (1992) showed that

(16)

with an analogous formula for the Lévy constant.

Real numbers  for which  include , , , and the golden ratio , plotted above.

Amazingly, the constant  is simply the limiting case  of a class of means defined by

(17)

for real  whose values are given by

(18)

(Ryll-Nardzewski 1951; Bailey et al. 1997; Khinchin 1997). An integral representation for  is given by

			(19)
			(20)

for , , ... (Iosifescu and Kraaikamp 2002, p. 231).

The constant

(21)

is sometimes known as the Khinchin harmonic mean and is the  case of an infinite family of such constants of which  and  are the first two members.

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

(22)

Then

(23)

for almost all real  (Khintchine 1934, 1936, Knuth 1981, Finch 2003), and

(24)

Furthermore, for , the limiting value

(25)

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

REFERENCES:

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

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.

Corless, R. M. "Continued Fractions and Chaos." Amer. Math. Monthly 99, 203-215, 1992.

Finch, S. R. "Khintchine-Lévy Constants." §1.8 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 59-65, 2003.

Gosper, R. W. "Simpler Khinchine [was: Re: my two cents]" math-fun@cs.arizona.edu mailing list. 25 Jun 1996.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 159, 2003.

Iosifescu, M. and Kraaikamp, C. Metrical Theory of Continued Fractions. Amsterdam, Netherlands: Kluwer, 2002.

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

Khinchin, A. Ya. "Average Values." §16 in Continued Fractions. New York: Dover, pp. 86-94, 1997.

Khintchine, A. "Metrische Kettenbruchprobleme." Compositio Math. 1, 361-382, 1934.

Khintchine, A. "Metrische Kettenbruchprobleme." Compositio Math. 2, 276-285, 1936.

Knuth, D. E. Exercise 24 in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 604, 1998.

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.

Lévy, P. "Sur les lois de probabilité dont dependent les quotients complets et incomplets d'une fraction continue." Bull. Soc. Math. France 57, 178-194, 1929.

Lévy, P. "Sur le développement en fraction continue d'un nombre choisi au hasard." Compositio Math. 3, 286-303, 1936. Reprinted in Œuvres de Paul Lévy, Vol. 6. Paris: Gauthier-Villars, pp. 285-302, 1980.

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

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

Ryll-Nardzewski, C. "On the Ergodic Theorems (I,II)." Studia Math. 12, 65-79, 1951.

Shanks, D. "Note MTE 164." Math. Tables Aids Comput. 4, 28, 1950.

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

Sloane, N. J. A. Sequences A002210/M1564, A002211/M0118, A086702, A087491, A087492, A087493, A087494, A087495, A087496, A087497, A087498, A087499, and A087500 in "The On-Line Encyclopedia of Integer Sequences."

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

Wrench, J. W. Jr. "Further Evaluation of Khinchin's Constant." Math. Comput. 14, 370-371, 1960.

Wrench, J. W. Jr. and Shanks, D. "Questions Concerning Khintchine's Constant and the Efficient Computation of Regular Continued Fractions." Math. Comput. 20, 444-448, 1966.