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Khinchin's Constant

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Let

x=[a_0;a_1,...]=a_0+1/(a_1+1/(a_2+1/(a_3+...)))
(1)

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

G_n(x)=(a_1a_2...a_n)^(1/n)
(2)

as n->infty. Amazingly, except for a set of measure 0, this limit is a constant independent of x given by

K=2.685452001...
(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 K is notoriously difficult to calculate to high precision, so computation of more digits get increasingly slower.

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

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

KhinchinsConstant

The values (a_1,a_2,...,a_n)^(1/n) are plotted above for n=1 to 500 and x=pi, sin1, the Euler-Mascheroni constant gamma, and the Copeland-Erdős constant C. Interestingly, the shape of the curves is almost identical to the corresponding curves for the Lévy constant.

If p_n/q_n is the nth convergent of the continued fraction of x, then

lim_(n->infty)q_n^(1/n)=lim_(n->infty)((p_n)/x)^(1/n)
(4)
=e^(pi^2/(12ln2))
(5)
=3.27582...
(6)

(OEIS A086702) for almost all real x (Lévy 1936, Finch 2003), where ln2 is the natural logarithm of 2. This number is sometimes called the Lévy constant.

Product expressions for K include

K=product_(n=1)^infty[1+1/(n(n+2))]^(lnn/ln2)
(7)

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

product_(k=1)^inftyk!^(Delta_k^3)lnk=K^(ln2),
(8)

where Delta_k^3 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 lnproduct_(k)a_k^(p_k)=sum_(k)p_klna_k. Sums for K include

K=exp[1/(ln2)sum_(m=1)^infty(H_(2m-1)^'[zeta(2m)-1])/m],
(9)

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

K=exp[1/(ln2)sum_(k=2)^infty((-1)^k(2-2^k)zeta^'(k))/k],
(10)

where zeta^'(z) 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

K=exp{-zeta^'(2,2)+1/(ln2)[sum_(k=2)^infty2(-1)^kf(k)]},
(11)

where

f(k)=(lnk)/((k+2)k^(k+2))[2^(k+1)_2F_1(1,k+2;k+3;-2/k)-_2F_1(1,k+2;k+3;-1/k)]+((2^k-1)zeta^'(k+1,k))/(k+1),
(12)

zeta(s,a) is a Hurwitz zeta function, zeta^'(s,a)=partialzeta/partials, and _2F_1(a,b;c;z) is a hypergeometric function.

Khinchin's constant is also given by the integrals

K=2exp{1/(ln2)int_0^11/(x(1+x))ln[(pix(1-x^2))/(sin(pix))]dx}
(13)
=2exp[1/(ln2)int_0^11/(x(1+x))ln[Gamma(2-x)Gamma(2+x)]dx]
(14)

(Shanks and Wrench 1959) and

K=exp[(pi^2)/(12ln2)+1/2ln2+1/(ln2)int_0^pi(ln(theta|cottheta|)dtheta)/theta].
(15)

Corless (1992) showed that

lnK=int_0^1(ln|_x^(-1)_|)/((x+1)ln2)dx,
(16)

with an analogous formula for the Lévy constant.

KhinchinsConstant2

Real numbers x for which lim_(n->infty)G_n(x)!=K include x=e, sqrt(2), sqrt(3), and the golden ratio phi, plotted above.

Amazingly, the constant K is simply the limiting case K=K_0 of a class of means defined by

K_p=lim_(n->infty)((a_1^p+a_2^p+...+a_n^p)/n)^(1/p)
(17)

for real p<1 whose values are given by

K_p={sum_(k=1)^infty-k^plg[1-1/((k+1)^2)]}^(1/p)
(18)

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

K_p=[1/(ln2)int_0^1(|_1/t_|^p)/(t+1)dt]^(1/p)
(19)
=[1/(ln2)sum_(k=1)^(infty)k^pln(1+1/(k(k+2)))]^(1/p)
(20)

for p=-1, -2, ... (Iosifescu and Kraaikamp 2002, p. 231).

The constant

K_(-1)=lim_(n->infty)n/(a_1^(-1)+a_2^(-1)+...+a_n^(-1))
(21)

is sometimes known as the Khinchin harmonic mean and is the p=-1 case of an infinite family of such constants of which K=K_0 and K_(-1) are the first two members.

Define the following quantity in terms of the kth partial quotient q_k,

M(s,n,x)=(1/nsum_(k=1)^nq_k^s)^(1/s).
(22)

Then

lim_(n->infty)M(1,n,x)=infty
(23)

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

M(1,n,x)∼O(lnn).
(24)

Furthermore, for s<1, the limiting value

lim_(n->infty)M(s,n,x)=K(s)
(25)

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

See also

Continued Fraction, Convergent, Gauss-Kuzmin-Wirsing Constant, Khinchin's Constant Approximations, Khinchin's Constant Continued Fraction, Khinchin's Constant Digits, Khinchin Harmonic Mean, Lévy Constant, Lochs' Constant, Lochs' Theorem, Partial Denominator, Simple Continued Fraction

Related Wolfram sites

http://functions.wolfram.com/Constants/Khinchin/

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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.

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Khinchin's Constant

Cite this as:

Weisstein, Eric W. "Khinchin's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KhinchinsConstant.html

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