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Khinchin's constant

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From Wikipedia, the free encyclopedia

Mathematical constant in number theory

Geometric means of the continued fraction coefficients for

{\displaystyle \pi }

(red),

{\displaystyle \gamma }

(blue) and

{\displaystyle {\sqrt[{3}]{2}}}

(green), conjectured to converge to Khinchin's constant (black).

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

{\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots }}}}}}}}\;}

it is almost always true that

lim

{\displaystyle \lim _{n\rightarrow \infty }\left(a_{1}a_{2}...a_{n}\right)^{1/n}=K_{0}.}

The decimal value of Khinchin's constant is given by:

2.68545

20010

65306

44530

{\displaystyle K_{0}=2.68545\,20010\,65306\,44530\dots }

(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:

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 ⁠

{\displaystyle \varphi }

⁠;

The base of the natural logarithm e.

Khinchin is sometimes spelled Khintchine (the French transliteration of Russian Хинчин) in older mathematical literature.

Series expressions<br>[edit]

Khinchin's constant can be given by the following infinite product:

log

{\displaystyle K_{0}=\prod _{r=1}^{\infty }{\left(1+{1 \over r(r+2)}\right)}^{\log _{2}r}}

This implies:

ln

ln

log

{\displaystyle \ln K_{0}=\sum _{r=1}^{\infty }\ln {\left(1+{1 \over r(r+2)}\right)}{\log _{2}r}}

Khinchin's constant may also be expressed as a rational zeta series in the form[1]

ln

ln

{\displaystyle \ln K_{0}={\frac {1}{\ln 2}}\sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\sum _{k=1}^{2n-1}{\frac {(-1)^{k+1}}{k}}}

or, by peeling off terms in the series,

ln

ln

ln

ln

{\displaystyle \ln K_{0}={\frac {1}{\ln 2}}\left[-\sum _{k=2}^{N}\ln \left({\frac {k-1}{k}}\right)\ln \left({\frac {k+1}{k}}\right)+\sum _{n=1}^{\infty }{\frac {\zeta (2n,N+1)}{n}}\sum _{k=1}^{2n-1}{\frac {(-1)^{k+1}}{k}}\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

ln

Li

Li

{\displaystyle \ln {\frac {K_{0}}{2}}={\frac {1}{\ln 2}}\left[{\mbox{Li}}_{2}\left({\frac {-1}{2}}\right)+{\frac {1}{2}}\sum _{k=2}^{\infty }(-1)^{k}{\mbox{Li}}_{2}\left({\frac {4}{k^{2}}}\right)\right].}

Integrals<br>[edit]

There exist a number of integrals related to Khinchin's constant:[2]

log

ln

{\displaystyle \int _{0}^{1}{\frac {\log _{2}\lfloor x^{-1}\rfloor }{x+1}}dx=\ln {K_{0}}}

log

ln

ln

{\displaystyle \int _{0}^{1}{\frac {\log _{2}(\Gamma (2+x)\Gamma (2-x))}{x(x+1)}}dx=\ln K_{0}-\ln 2}

log

sin

ln

ln

{\displaystyle \int _{0}^{1}{\frac {1}{x(x+1)}}\log _{2}\left({\frac {\pi x(1-x^{2})}{\sin \pi x}}\right)dx=\ln K_{0}-\ln 2}

log

cot

ln

ln

12<br>ln

{\displaystyle \int _{0}^{\pi }{\frac {\log _{2}(x|\cot x|)}{x}}dx=\ln K_{0}-{\frac {1}{2}}\ln 2-{\frac {\pi ^{2}}{12\ln 2}}}

Sketch of proof<br>[edit]

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

{\displaystyle I=[0,1]\setminus \mathbb {Q} }

. 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

{\displaystyle T([a_{1},a_{2},\dots ])=[a_{2},a_{3},\dots ].\,}

The transformation...