|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[number theory]], [[Aleksandr Yakovlevich Khinchin]] proved that for [[almost all]] real numbers ''x'', coefficients ''a''<sub>''i''</sub> of the [[continued fraction]] expansion of ''x'' have a finite [[geometric mean]] that is independent of the value of ''x'' and is known as '''Khinchin's constant'''.
| | Gabrielle Straub is what for you can call me although it's not the numerous feminine of names. Guam has always been my house. Filing has been my profession smoothly time and I'm doing pretty good financially. Fish keeping is what I follow every week. See what precisely new on my web presence here: http://circuspartypanama.com<br><br>Also visit my blog; [http://circuspartypanama.com clash of clans hack tool no survey download] |
| | |
| That is, for
| |
| | |
| :<math>x = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cfrac{1}{\ddots}}}}\;</math>
| |
| | |
| it is [[almost all|almost always]] true that | |
| | |
| :<math>\lim_{n \rightarrow \infty } \left( \prod_{i=1}^n a_i \right) ^{1/n} =
| |
| K_0</math>
| |
| where <math>\prod</math> is [[Multiplication#Capital_Pi_notation|sequence multiplication]] <math>( a_1 * a_2 * ... a_n )</math> and <math>K_0</math> is Khinchin's constant
| |
| :<math>K_0 =
| |
| \prod_{r=1}^\infty {\left( 1+{1\over r(r+2)}\right)}^{\log_2 r} \approx 2.6854520010\dots</math> {{OEIS|id=A002210}}.
| |
| | |
| Among the numbers ''x'' whose continued fraction expansions do ''not'' have this property are [[rational number]]s, solutions of [[quadratic equation]]s with rational coefficients (including the [[golden ratio]] Φ), and the [[e (mathematical constant)|base of the natural logarithm]] ''e''.
| |
| | |
| Khinchin is sometimes spelled Khintchine (the French transliteration of Russian Хи́нчин) in older mathematical literature.
| |
| | |
| ==Sketch of proof==
| |
| The proof presented here was arranged by {{harvtxt|Ryll-Nardzewski|1951}} and is much simpler than Khinchin's original proof which did not use [[ergodic theory]].
| |
| | |
| Since the first coefficient ''a''<sub>0</sub> 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 <math>\scriptstyle I=[0,1]\setminus\mathbb{Q}</math>. These numbers are in [[bijection]] with infinite [[continued fraction]]s of the form [0; ''a''<sub>1</sub>, ''a''<sub>2</sub>, ...], which we simply write [''a''<sub>1</sub>, ''a''<sub>2</sub>, ...], where ''a''<sub>1</sub>, ''a''<sub>2</sub>, ... are [[positive integer]]s. Define a transformation ''T'':''I'' → ''I'' by
| |
| | |
| :<math>T([a_1,a_2,\dots])=[a_2,a_3,\dots].\,</math>
| |
| | |
| The transformation ''T'' is called the [[Gauss–Kuzmin–Wirsing operator]]. For every [[Borel set|Borel subset]] ''E'' of ''I'', we also define the [[Gauss–Kuzmin distribution|Gauss–Kuzmin measure]] of ''E''
| |
| | |
| :<math>\mu(E)=\frac{1}{\log 2}\int_E\frac{dx}{1+x}.</math>
| |
| | |
| Then ''μ'' is a [[probability measure]] on the [[Sigma-algebra|''σ''-algebra]] of Borel subsets of ''I''. The measure ''μ'' is [[Equivalence (measure theory)|equivalent]] to the Lebesgue measure on ''I'', but it has the additional property that the transformation ''T'' [[measure-preserving transformation|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 <math>f \left( T^k x \right)</math> is the same for almost all <math>x</math>:
| |
| | |
| :<math>\lim_{n\to\infty} \frac 1n\sum_{k=0}^{n-1}(f\circ T^k)(x)=\int_I f d\mu\quad\text{for }\mu\text{-almost all }x\in I.</math> | |
| | |
| Applying this to the function defined by ''f''([''a''<sub>1</sub>, ''a''<sub>2</sub>, ...]) = log(''a''<sub>1</sub>), we obtain that
| |
| | |
| :<math>\lim_{n\to\infty}\frac 1n\sum_{k=1}^{n}\log(a_k)=\int_I f \, d\mu = \sum_{r=1}^\infty\log(r)\frac{\log\bigl(1+\frac{1}{r(r+2)}\bigr)}{\log 2}</math>
| |
| | |
| for almost all [''a''<sub>1</sub>, ''a''<sub>2</sub>, ...] in ''I'' as ''n'' → ∞.
| |
| | |
| Taking the [[exponential function|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.
| |
| | |
| ==Series expressions==
| |
| Khinchin's constant may be expressed as a [[rational zeta series]] in the form
| |
| | |
| :<math>\log K_0 = \frac{1}{\log 2} \sum_{n=1}^\infty
| |
| \frac {\zeta (2n)-1}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k}
| |
| </math>
| |
| or, by peeling off terms in the series,
| |
| :<math>\log K_0 = \frac{1}{\log 2} \left[
| |
| \sum_{k=3}^N \log \left(\frac{k-1}{k} \right) \log \left(\frac{k+1}{k} \right)
| |
| + \sum_{n=1}^\infty
| |
| \frac {\zeta (2n,N)}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k}
| |
| \right]
| |
| </math> | |
| | |
| where ''N'' is an integer, held fixed, and ζ(''s'', ''n'') is the [[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]]:
| |
| | |
| :<math>\log K_0 = \log 2 + \frac{1}{\log 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].
| |
| </math>
| |
| | |
| ==Hölder means==
| |
| The Khinchin constant can be viewed as the first in a series of the [[Hölder mean]]s of the terms of continued fractions. Given an arbitrary series {''a''<sub>''n''</sub>}, the Hölder mean of order ''p'' of the series is given by
| |
| | |
| :<math>K_p=\lim_{n\to\infty} \left[\frac{1}{n}
| |
| \sum_{k=1}^n a_k^p \right]^{1/p}.</math>
| |
| | |
| When the {''a''<sub>''n''</sub>} are the terms of a continued fraction expansion, the constants are given by
| |
| | |
| :<math>K_p=\left[\sum_{k=1}^\infty -k^p
| |
| \log_2\left( 1-\frac{1}{(k+1)^2} \right)
| |
| \right]^{1/p}.</math>
| |
| | |
| This is obtained by taking the ''p''-th mean in conjunction with the [[Gauss–Kuzmin distribution]]. The value for ''K''<sub>0</sub> may be shown to be obtained in the limit of ''p'' → 0.
| |
| | |
| ==Harmonic mean==
| |
| By means of the above expressions, the [[harmonic mean]] of the terms of a continued fraction may be obtained as well. The value obtained is
| |
| | |
| :<math>K_{-1}=1.74540566240\dots</math> {{OEIS|A087491}}.
| |
| | |
| ==Open problems==
| |
| * Among the numbers whose geometric mean of the coefficients ''a''<sub>''i''</sub> in the continued fraction expansion apparently (based on numerical evidence) tends to Khinchin's constant are [[pi|{{pi}}]], the [[Euler–Mascheroni constant]] γ, and Khinchin's constant itself. However, none of these limits has been rigorously established, even though it is known that [[almost all]] real numbers have this property.
| |
| * It is not known if Khinchin's constant is a rational, [[Algebraic numbers|algebraic]] [[Irrational numbers|irrational]] or [[Transcendental numbers|transcendental]] number.<ref>{{MathWorld|urlname=KhinchinsConstant|title=Khinchin's constant}}</ref>
| |
| | |
| ==See also==
| |
| * [[Lévy's constant]]
| |
| | |
| ==References==
| |
| * {{cite journal|author=David H. Bailey, Jonathan M. Borwein, Richard E. Crandall
| |
| |url=http://www.reed.edu/~crandall/papers/95-036-Bailey-Borwein-Crandall.pdf
| |
| |title=On the Khinchine constant
| |
| |journal=
| |
| |year=1995
| |
| |volume=
| |
| |pages=
| |
| }}
| |
| | |
| * {{cite journal|author=Jonathan M. Borwein, David M. Bradley, Richard E. Crandall
| |
| |url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf
| |
| |title=Computational Strategies for the Riemann Zeta Function
| |
| |journal=J. Comp. App. Math.
| |
| |year=2000
| |
| |volume=121
| |
| |pages=p.11
| |
| }}
| |
| | |
| * {{cite book|author=Aleksandr Ya. Khinchin|title=Continued Fractions|publisher=Dover Publications|location=New York|year=1997}}
| |
| | |
| * {{citation|last=Ryll-Nardzewski|first=Czesław|title=On the ergodic theorems II (Ergodic theory of continued fractions)|journal=Studia Mathematica|volume=12|year=1951|pages=74–79}}
| |
| | |
| <references/>
| |
| | |
| ==External links==
| |
| * [http://pi.lacim.uqam.ca/piDATA/khintchine.txt 110,000 digits of Khinchin's constant]
| |
| * [http://mpmath.googlecode.com/svn/data/khinchin.txt 10,000 digits of Khinchin's constant]
| |
| | |
| [[Category:Continued fractions]]
| |
| [[Category:Mathematical constants]]
| |
Gabrielle Straub is what for you can call me although it's not the numerous feminine of names. Guam has always been my house. Filing has been my profession smoothly time and I'm doing pretty good financially. Fish keeping is what I follow every week. See what precisely new on my web presence here: http://circuspartypanama.com
Also visit my blog; clash of clans hack tool no survey download