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 ''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

:$x = a_0+\cfrac\;$

it is almost always true that

:$\lim_ \left( a_1 a_2 ... a_n \right) ^ = K_0.$
The decimal value of Khinchin's constant is given by:
:$K_0 = 2.68545\, 20010 \, 65306\, 44530 \dots$

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

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 (however, the geometric mean of all coefficients for square roots of nonsquare integers from 2 to 24 is about 2.708, suggesting that quadratic roots collectively may give the Khinchin constant as a geometric mean);
* 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:

:$K_0=\prod_^\infty ^$

This implies:

:$\ln K_0=\sum_^\infty \ln$

Khinchin's constant may also be expressed as a rational zeta series in the form

:$\ln K_0 = \frac \sum_^\infty \frac \sum_^ \frac$
or, by peeling off terms in the series,
:\ln K_0 = \frac \left -\sum_^N \ln \left(\frac \right) \ln \left(\frac \right)
+ \sum_^\infty
\frac \sum_^ \frac
\right

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:

:\ln \frac = \frac \left \mbox_2 \left( \frac \right) +
\frac\sum_^\infty (-1)^k \mbox_2 \left( \frac \right)
\right

Integrals

There exist a number of integrals related to Khinchin's constant:

: $\int_0^1 \frac \mathrm dx = \ln$

: $\int_0^1 \frac \mathrm dx = \ln K_0-\ln2$

: $\int_0^1 \frac\log_2\left(\frac\right) \mathrm dx = \ln K_0 - \ln2$

: $\int_0^\pi \frac\mathrm dx = \ln K_0 - \frac12\ln2 - \frac$

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 I= ,1setminus\mathbb. These numbers are in bijection with infinite continued fractions of the form ; ''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

:T( _1,a_2,\dots= _2,a_3,\dots\,

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''

:$\mu(E)=\frac\int_E\frac.$

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 $f \left( T^k x \right)$ is the same for almost all $x$:

:$\lim_ \frac 1n\sum_^(f\circ T^k)(x)=\int_I f d\mu\quad\text\mu\textx\in I.$

Applying this to the function defined by ''f''( 'a''₁, ''a''₂, ... = ln(''a''₁), we obtain that

:\lim_\frac 1n\sum_^\ln a_k=\int_I f \, d\mu = \sum_^\infty\ln\left +\frac\rightlog_2r

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 , the Hölder mean of order ''p'' of the series is given by

:K_p=\lim_ \left frac
\sum_^n a_k^p \right.
When the are the terms of a continued fraction expansion, the constants are given by
:K_p=\left sum_^\infty -k^p
\log_2\left( 1-\frac \right)
\right.

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: $\lim_\frac 1n \sum_^n a_k = K_1 = +\infty$, and so the coefficients grow arbitrarily large: $\limsup_n a_n = +\infty$.

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

The harmonic mean (''p'' = −1) is
:$K_=1.74540566240\dots$ .

Open problems

Many well known numbers, such as , 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 $\lim_ \left( a_1 a_2 ... a_n \right) ^$ 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.

See also

* Lochs' theorem
* Lévy's constant
* Somos' constant
* List of mathematical constants

References

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External links

{{Commons category, Khinchin's constant

110,000 digits of Khinchin's constant

10,000 digits of Khinchin's constant

Continued fractions
Mathematical constants
Infinite products