Mathematical constant in number theory
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 "a""i" of the continued fraction expansion of "x" have a finite geometric mean that is independent of the value of "x." It is known as Khinchin's constant and denoted by "K0."
That is, for
formula_1
it is almost always true that
formula_2
The decimal value of Khinchin's constant is given by:
formula_3 (sequence in the OEIS)
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:
However, not a single real number "x" has been verified to have this property.
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.
Series expressions.
Khinchin's constant can be given by the following infinite product:
formula_4
This implies:
formula_5
Khinchin's constant may also be expressed as a rational zeta series in the form
formula_6
or, by peeling off terms in the series, 
formula_7
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:
formula_8
Integrals.
There exist a number of integrals related to Khinchin's constant:
 formula_9
 formula_10
 formula_11
 formula_12
Sketch of proof.
The proof presented here was arranged by Czesław Ryll-Nardzewski and is much simpler than Khinchin's original proof which did not use ergodic theory.
Since the first coefficient "a"0 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 formula_13. These numbers are in bijection with infinite continued fractions of the form [0; "a"1, "a"2, ...], which we simply write ["a"1, "a"2, ...], where "a"1, "a"2, ... are positive integers. Define a transformation "T":"I" → "I" by
formula_14
The transformation "T" is called the Gauss–Kuzmin–Wirsing operator. For every Borel subset "E" of "I", we also define the Gauss–Kuzmin measure of "E"
formula_15
Then "μ" is a probability measure on the "σ"-algebra of Borel subsets of "I". The measure "μ" is equivalent to the Lebesgue measure on "I", but it has the additional property that the transformation "T" preserves the measure "μ". Moreover, it can be proved that "T" is an ergodic transformation of the measurable space "I" endowed with the probability measure "μ" (this is the hard part of the proof). The ergodic theorem then says that for any "μ"-integrable function "f" on "I", the average value of formula_16 is the same for almost all formula_17:
formula_18
Applying this to the function defined by "f"(["a"1, "a"2, ...]) = ln("a"1), we obtain that
formula_19
for almost all ["a"1, "a"2, ...] 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.
Generalizations.
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 {"a""n"}, the Hölder mean of order "p" of the series is given by
formula_20
When the {"a""n"} are the terms of a continued fraction expansion, the constants are given by
formula_21
This is obtained by taking the "p"-th mean in conjunction with the Gauss–Kuzmin distribution. This is finite when formula_22.
The arithmetic average diverges: formula_23, and so the coefficients grow arbitrarily large: formula_24.
The value for "K"0 is obtained in the limit of "p" → 0.
The harmonic mean ("p" = −1) is
formula_25 (sequence in the OEIS).
Open problems.
Many well known numbers, such as π, the Euler–Mascheroni constant γ, and Khinchin's constant itself, based on numerical evidence, are thought to be among the numbers for which the limit formula_26 converges to Khinchin's constant. However, none of these limits have been rigorously established. In fact, it has not been proven for "any" real number, which was not specifically constructed for that exact purpose.
The algebraic properties of Khinchin's constant itself, e. g. whether it is a rational, algebraic irrational, or transcendental number, are also not known.