辛钦常数

以下是x的连分数表示式 （公式略）

针对任意实数x，以下的等式几乎总是为真 lim _(n→ ∞)(∏ ᵢ₌₁ⁿaᵢ)^(1/n)=K₀

其中 K₀ 为辛钦常数 （公式略） （OEIS数列A002210）.

不符合上述条件的实数包括了有理数、实系数二次方程的解（包括黄金比例 （公式略）），以及自然对数的底e。目前辛钦常数是否为无理数或代数数仍犹未可知。虽然几乎所有实数之连分数系数的几何平均都趋近于辛钦常数，但除了特意建构的实数外，并没有实数被严格证明有此性质，仅有一些数值上的证据，像是圆周率及欧拉-马歇罗尼常数。

开放问题

根据数值上的证据，圆周率π、欧拉-马斯刻若尼常数γ、以及辛钦常数本身的连分数系数ai的几何平均数会趋近辛钦常数，不过这还没有严谨的证明。

目前还不知道辛钦常数是有理数、代数数、无理数或超越数。

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. It is known as Khinchin's constant and denoted by K0.

That is, for （公式略）

it is almost always true that lim _(n→ ∞)(a₁a₂...aₙ)^(1/n)=K₀.

The decimal value of Khinchin's constant is given by: K₀=2.68545 20010 65306 44530… (sequence A002210 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:

• π
• Roots of equations with a degree > 2, e.g. cubic roots and quartic roots
• Natural logarithms, e.g. ln(2) and ln(3)
• The Euler-Mascheroni constant γ
• Apéry's constant ζ(3)
• The Feigenbaum constants δ and α
• Khinchin's constant itself (which would mean it is irrational)

Among the numbers x whose continued fraction expansions are known not to have this property are:

Rational numbers

Roots of quadratic equations, e.g. the square roots of integers and the golden ratio ⁠ φ ⁠;

The base of the natural logarithm e.

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: （公式略）
• This implies: （公式略）
• Khinchin's constant may also be expressed as a rational zeta series in the form （公式略）
• or, by peeling off terms in the series, （公式略）

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: （公式略）

Integrals

• There exist a number of integrals related to Khinchin's constant: （公式略）
• ∫ （公式略）
• ∫ （公式略）
• ∫ （公式略）

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 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 （公式略）. 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. Define a transformation T:I → I by T([a₁,a₂,… ])=[a₂,a₃,… ].

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 μ (E)=1/(ln 2)∫ _(E)dx/(1+x).

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).

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 {an}, the Hölder mean of order p of the series is given by Kₚ=lim _(n→ ∞)[1/n∑ ₖ₌₁ⁿaₖ^(p)]^(1/p).

When the {an} are the terms of a continued fraction expansion, the constants are given by （公式略）

This is obtained by taking the p-th mean in conjunction with the Gauss–Kuzmin distribution. This is finite when p<1.

The arithmetic average diverges: lim _(n→ ∞)1/n∑ ₖ₌₁ⁿaₖ=K₁=+∞, and so the coefficients grow arbitrarily large: （公式略）.

The value for K0 is obtained in the limit of p → 0.

The harmonic mean (p = −1) is K₋₁=1.74540566240… (sequence A087491 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 lim _(n→ ∞)(a₁a₂...aₙ)^(1/n) converges to Khinchin's constant. However, none of these limits has 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.