Khinchin's constant

In number theory, Khinchin's constant is a mathematical constant related to the simple continued fraction expansions of many real numbers.

In particular Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x.

That is, for it is almost always true that The decimal value of Khinchin's constant is given by: Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose.

The following numbers whose continued fraction expansions apparently do have this property (based on empirical data) are: Among the numbers x whose continued fraction expansions are known not to have this property are: Khinchin is sometimes spelled Khintchine (the French transliteration of Russian Хинчин) in older mathematical literature.

Khinchin's constant can be given by the following infinite product: This implies: Khinchin's constant may also be expressed as a rational zeta series in the form[1] or, by peeling off terms in the series, where N is an integer, held fixed, and ζ(s, n) is the complex Hurwitz zeta function.

Both series are strongly convergent, as ζ(n) − 1 approaches zero quickly for large n. An expansion may also be given in terms of the dilogarithm: There exist a number of integrals related to Khinchin's constant:[2] The proof presented here was arranged by Czesław Ryll-Nardzewski[3] and is much simpler than Khinchin's original proof which did not use ergodic theory.

Since the first coefficient a0 of the continued fraction of x plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure zero, we are reduced to the study of irrational numbers in the unit interval, i.e., those in

These numbers are in bijection with infinite continued fractions of the form [0; a1, a2, ...], which we simply write [a1, a2, ...], where a1, a2, ... are positive integers.

The ergodic theorem then says that for any μ-integrable function f on I, the average value of

: Applying this to the function defined by f([a1, a2, ...]) = ln(a1), we obtain that for almost all [a1, a2, ...] in I as n → ∞.

Taking the exponential on both sides, we obtain to the left the geometric mean of the first n coefficients of the continued fraction, and to the right Khinchin's constant.

The Khinchin constant can be viewed as the first in a series of the Hölder means of the terms of continued fractions.

Given an arbitrary series {an}, the Hölder mean of order p of the series is given by When the {an} are the terms of a continued fraction expansion, the constants are given by This is obtained by taking the p-th mean in conjunction with the Gauss–Kuzmin distribution.

The harmonic mean (p = −1) is Many well known numbers, such as π, the Euler–Mascheroni constant γ, and Khinchin's constant itself, based on numerical evidence,[4][5][2] are thought to be among the numbers for which the limit

In fact, it has not been proven for any real number, which was not specifically constructed for that exact purpose.