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). This number is sometimes called the Lévy constant.

Product and sum expressions for include

(7)

(Shanks and Wrench 1959; Khinchin 1997, p. 93; Borwein and Bailey 2003, p. 25; Havil 2003, p. 161) and

(8)

where is the Riemann zeta function and is a an alternating harmonic number (Bailey et al. 1997). Gosper (pers. comm., Jun. 25, 1996) gave

(9)

where is the derivative of the Riemann zeta function. An 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

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

(10)

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),

(11)

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

Khinchin's constant is also given by the integrals

		$2exp{1/(ln2)int_0^11/(x(1+x))ln[(pix(1-x^2))/(sin(pix))]dx}$	(12)
			(13)

(Shanks and Wrench 1959) and

(14)

Corless (1992) showed that

(15)

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

(16)

for real whose values are given by

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

(17)

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

			(18)
			(19)

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

The constant

(20)

is sometimes known as the Khinchin harmonic mean. The following table summarizes the values for the first few nonpositive integers (Bailey et al. 1997, Plouffe).

	Sloane	value
0	A002210	2.685452001065306445309714835481795693820382293994462
	A087491	1.745405662407346863494596309683661067294936618777984
	A087492	1.450340328495630406052983076680697881408299979605904
	A087493	1.313507078687985766717339447072786828158129861484792
	A087494	1.236961809423730052626227244453422567420241131548937
	A087495	1.189003926465513154062363732771403397386092512639671
	A087496	1.156552374421514423152605998743410046840213070718761
	A087497	1.133323363950865794910289694908868363599098282411797
	A087498	1.115964408978716690619156419345349695769491182230400
	A087499	1.102543136670728013836093402522568351022221284149318
	A087500	1.091877041209612678276110979477638256493272651429656