Khintchine's Constant

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

Let

$$x=[a_0, a_1, \ldots]=a_0+{1\over\strut\displaystyle a_1+{\st... ...2+{\strut\displaystyle 1\over\strut\displaystyle a_3+\ldots}}}$$ (1)

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

$$G_n(x)=(a_1a_2\cdots a_n)^{1/n}$$ (2)

as $n\to\infty$. Amazingly enough, this limit is a constant independent of $x$--except if $x$ belongs to a set of Measure 0-given by

$$K=2.685452001\ldots$$ (3)

(Sloane's A002210), as proved in Kac (1959). The values $G_n(x)$ are plotted above for $n=1$ to 500 and $x=\pi$, $1/\pi$, $\sin 1$, the Euler-Mascheroni Constant $\gamma$, and the Copeland-Erdös Constant. Real Numbers $x$ for which $\lim_{n\to\infty} G_n(x)\not=K$ include $x=e$, $\sqrt{2}$, $\sqrt{3}$, and the Golden Ratio $\phi$, plotted below.

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

Explicit expressions for $K$ include

$$K = \prod_{n=1}^\infty \left[{1+{1\over n(n+2)}}\right]^{\ln n/\ln 2}$$ (4)

$$\ln 2\ln K = {\textstyle{1\over 12}}\pi^2+{\textstyle{1\over 2}}(\ln 2)^2+\int_0^\pi {\ln(\theta\vert\cot\theta\vert)\,d\theta\over\theta}$$ (5)

$$\ln K = {1\over\ln 2}\sum_{m=1}^\infty {h_{m-1}\over m}[\zeta(2m)-1],$$ (6)

where $\zeta(z)$ is the Riemann Zeta Function and

$$h_m=\sum_{j=1}^m {(-1)^{j-1}\over j}$$ (7)

(Shanks and Wrench 1959). Gosper gave

$$\ln K = {1\over\ln 2} \sum_{j=2}^\infty {(-1)^j(2-2^j)\zeta'(j)\over j},$$ (8)

where $\zeta'(z)$ is the Derivative of the Riemann Zeta Function. An extremely rapidly converging sum also due to Gosper is

$\ln K={1\over\ln 2} \sum_{k=0}^\infty \Biggl\{-\ln(k+1)[\ln(k+3)-2\ln(k+2)+\ln(k+1)]$
$-{(-1)^k(2-2^{k+2})\over k+2} \left[{{\ln(k+1)\over (k+1)^{k+2}}-\zeta'(k+2,k+... ...+1)\left[{\sum_{s=1}^{k+2} {(-1)^s(2-2^s)\over(k+1)^s s}}\right]}\right\},\quad$	(9)

where $\zeta(s,a)$ is the Hurwitz Zeta Function.

Khintchine's constant is also given by the integral

$$\ln 2\ln({\textstyle{1\over 2}}K)=\int_0^1 {1\over x(1+x)} \ln\left[{\pi x(1-x^2)\over\sin(\pi x)}\right]\,dx.$$ (10)

If $P_n/Q_n$ is the $n$th Convergent of the Continued Fraction of $x$, then

$$\lim_{n\to\infty} (Q_n)^{1/n}=\lim_{n\to\infty} \left({P_n\over x}\right)^{1/n} = e^{\pi^2/(12\ln 2)}\approx 3.27582$$ (11)

for almost all Real $x$ (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 $k$th partial quotient $q_k$,

$$M(s,n,x)=\left({{1\over n} \sum_{k=1}^n {q_k}^s}\right)^{1/s}.$$ (12)

Then

$$\lim_{n\to\infty} M(1,n,x)=\infty$$ (13)

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

$$M(1,n,x)\sim{\mathcal O}(\ln n).$$ (14)

Furthermore, for $s<1$, the limiting value

$$\lim_{n\to\infty}M(s,n,x)=K(s)$$ (15)

exists and is a constant $K(s)$ 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.