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).