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Khinchin's constant
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{{Short description|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 number|real numbers]]. In particular [[Aleksandr Yakovlevich Khinchin]] proved that for [[almost all]] real numbers ''x'', the 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.'' It is known as Khinchin's constant and denoted by ''K<sub>0</sub>.'' 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( a_1 a_2 ... a_n \right) ^{1/n} = K_0.</math> The decimal value of Khinchin's constant is given by: :<math>K_0 = 2.68545\, 20010 \, 65306\, 44530 \dots</math> {{OEIS|id=A002210}} 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: * [[Pi |π]] * Roots of equations with a degree > 2, ''e.g.'' [[Cubic equation|cubic]] roots and [[Quartic equation|quartic]] roots * [[Natural logarithm|Natural logarithms]], ''e.g.'' [[Natural logarithm of 2|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 number]]s * Roots of [[quadratic equation]]s, ''e.g.'' the [[square root]]s of integers and the [[golden ratio]] {{tmath|\varphi}} (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 [[e (mathematical constant)|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: :<math>K_0=\prod_{r=1}^\infty {\left( 1+{1\over r(r+2)}\right)}^{\log_2 r}</math> This implies: :<math>\ln K_0=\sum_{r=1}^\infty \ln{\left( 1+{1\over r(r+2)}\right)}{\log_2 r}</math> Khinchin's constant may also be expressed as a [[rational zeta series]] in the form<ref>Bailey, Borwein & Crandall, 1997. In that paper, a slightly non-standard definition is used for the Hurwitz zeta function.</ref> :<math>\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} </math> or, by peeling off terms in the series, :<math>\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] </math> 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]]: :<math>\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]. </math> ==Integrals== There exist a number of integrals related to Khinchin's constant:<ref name=":0">{{MathWorld|urlname=KhinchinsConstant|title=Khinchin's constant}}</ref> : <math> \int_0^1 \frac{\log_2 \lfloor x^{-1} \rfloor}{x+1} \mathrm dx = \ln {K_0} </math> : <math> \int_0^1 \frac{\log_2(\Gamma(2+x)\Gamma(2-x))}{x(x+1)} \mathrm dx = \ln K_0-\ln2 </math> : <math> \int_0^1 \frac{1}{x(x+1)}\log_2\left(\frac{\pi x (1-x^2)}{\sin \pi x}\right) \mathrm dx = \ln K_0 - \ln2 </math> : <math> \int_0^\pi \frac{\log_2(x|\cot x|)}{x}\mathrm dx = \ln K_0 - \frac12\ln2 - \frac{\pi^2}{12\ln 2} </math> ==Sketch of proof== The proof presented here was arranged by [[Czesław Ryll-Nardzewski]]<ref>{{Cite journal |last=Ryll-Nardzewski |first=C. |date=1951 |title=On the ergodic theorems (II) (Ergodic theory of continued fractions) |url=http://www.impan.pl/get/doi/10.4064/sm-12-1-74-79 |journal=Studia Mathematica |language=en |volume=12 |issue=1 |pages=74–79 |doi=10.4064/sm-12-1-74-79 |issn=0039-3223|url-access=subscription }}</ref> 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>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}{\ln 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>, ...]) = ln(''a''<sub>1</sub>), we obtain that :<math>\lim_{n\to\infty}\frac 1n\sum_{k=1}^{n}\ln a_k=\int_I f \, d\mu = \sum_{r=1}^\infty\ln\left[1+\frac{1}{r(r+2)}\right]\log_2r</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. ==Generalizations== 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]]. This is finite when <math>p < 1</math>. The arithmetic average diverges: <math>\lim_{n\to\infty}\frac 1n \sum_{k=1}^n a_k = K_1 = +\infty</math>, and so the coefficients grow arbitrarily large: <math>\limsup_n a_n = +\infty</math>. The value for ''K''<sub>0</sub> is obtained in the limit of ''p'' → 0. The [[harmonic mean]] (''p'' = −1) is :<math>K_{-1}=1.74540566240\dots</math> {{OEIS|A087491}}. ==Open problems== [[File:Geometric means of continued fractions to Khinchin constant.svg|thumb|The limits for <math>\sin1</math> (green), <math>e</math> (red), <math>\sqrt{31}</math> (blue) and a constructed number (yellow).]] Many well known numbers, such as [[pi|{{pi}}]], the [[Euler–Mascheroni constant]] γ, and Khinchin's constant itself, based on numerical evidence,<ref>{{Cite web|url=https://mathworld.wolfram.com/Euler-MascheroniConstantContinuedFraction.html|title=Euler-Mascheroni Constant Continued Fraction|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2020-03-23}}</ref><ref>{{Cite web|url=https://mathworld.wolfram.com/PiContinuedFraction.html|title=Pi Continued Fraction|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2020-03-23}}</ref><ref name=":0" /> are thought to be among the numbers for which the limit <math>\lim_{n \rightarrow \infty } \left( a_1 a_2 ... a_n \right) ^{1/n}</math> 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.<ref>{{Cite journal |last=Wieting |first=Thomas |date=2008 |title=A Khinchin Sequence |url=https://www.ams.org/journals/proc/2008-136-03/S0002-9939-07-09202-7/ |journal=Proceedings of the American Mathematical Society |language=en |volume=136 |issue=3 |pages=815–824 |doi=10.1090/S0002-9939-07-09202-7 |issn=0002-9939|doi-access=free }}</ref> The algebraic properties of Khinchin's constant itself, e. g. whether it is a rational, [[Algebraic numbers|algebraic]] [[Irrational numbers|irrational]], or [[Transcendental numbers|transcendental]] number, are also not known.<ref name=":0" /> ==See also== * [[Lochs' theorem]] * [[Lévy's constant]] * [[Somos' quadratic recurrence constant|Somos' constant]] * [[List of mathematical constants]] ==References== <references/> * {{cite journal|year=1995|title=On the Khinchine constant|url=http://www.davidhbailey.com/dhbpapers/khinchine.pdf|doi=10.1090/s0025-5718-97-00800-4|author1=David H. Bailey|author2=Jonathan M. Borwein|author3=Richard E. Crandall|journal=Mathematics of Computation|volume=66|issue=217|pages=417–432|doi-access=free}} * {{cite journal|author1=Jonathan M. Borwein |author2=David M. Bradley |author3=Richard E. Crandall |url=http://www.maths.ex.ac.uk/~mwatkins/zeta/borwein1.pdf |title=Computational Strategies for the Riemann Zeta Function |journal=J. Comput. Appl. Math. |year=2000 |volume=121 |issue=1–2 |page=11 |doi=10.1016/s0377-0427(00)00336-8 |bibcode=2000JCoAM.121..247B |doi-access=free }} * {{cite journal|author=Thomas Wieting |url=https://www.ams.org/journals/proc/2008-136-03/S0002-9939-07-09202-7/ |title=A Khinchin Sequence |journal=Proceedings of the American Mathematical Society |year=2007 |volume=136 |issue=3 |pages=815–824 |doi=10.1090/S0002-9939-07-09202-7 |doi-access=free }} * {{cite book|author=Aleksandr Ya. Khinchin|title=Continued Fractions|publisher=Dover Publications|location=New York|year=1997}} ==External links== {{Commons category|Khinchin's constant}} * [http://www.plouffe.fr/simon/constants/khintchine.txt 110,000 digits of Khinchin's constant] * [https://web.archive.org/web/20081101100001/http://mpmath.googlecode.com/svn/data/khinchin.txt 10,000 digits of Khinchin's constant] [[Category:Continued fractions]] [[Category:Mathematical constants]] [[Category:Infinite products]]
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