Khinchin's constant

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In number theory, Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the infinitely many denominators a_i of the continued fraction expansion of x have an astonishing property: their geometric mean is a constant, known as Khinchin's constant, which is independent of the value of x.

That is, for

$x = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\ldots}}}\;$

it is almost always true that

$\lim_{n \rightarrow \infty } \left( \prod_{i=1}^n a_i \right) ^{1/n} = K_0$

where $K 0$ is Khinchin's constant

$K_0 = \prod_{r=1}^\infty {\left\{ 1+{1\over r(r+2)}\right\}}^{\log_2 r} \approx 2.6854520010\dots$

Among the numbers x whose continued fraction expansions do not have this property are rational numbers, solutions of quadratic equations with rational coefficients (including the golden ratio φ), and the base of the natural logarithms e.

Among the numbers whose continued fraction expansions apparently do have this property (based on numerical evidence) are π, the Euler-Mascheroni constant γ, and Khinchin's constant itself. However this is unproven, because even though almost all real numbers are known to have this property, it has not been proven for any specific real number whose full continued fraction representation is not known.

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

1 Series expressions
2 Hölder means
3 Harmonic mean
4 See also
5 References

[edit] Series expressions

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

$\log K_0 = \frac{1}{\log 2} \sum_{n=1}^\infty \frac {\zeta (2n)-1}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k}$

or, by peeling off terms in the series,

$\log K_0 = \frac{1}{\log 2} \left[ \sum_{k=3}^N \log \left(\frac{k-1}{k} \right) \log \left(\frac{k+1}{k} \right) + \sum_{n=1}^\infty \frac {\zeta (2n,N)}{n} \sum_{k=1}^{2n-1} \frac{(-1)^{k+1}}{k} \right]$

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:

$\log K_0 = \log 2 + \frac{1}{\log 2} \left[ \mbox{Li}_2 \left( \frac{-1}{2} \right) + \frac{1}{2}\sum_{k=2}^\infty (-1)^k \mbox{Li}_2 \left( \frac{4}{k^2} \right) \right]$

[edit] Hölder means

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 ${a n}$ , the Hölder mean of order p of the series is given by

$K_p=\lim_{n\to\infty} \left[\frac{1}{n} \sum_{k=1}^n a_k^p \right]^{1/p}$

When the ${a n}$ are the terms of a continued fraction expansion, the constants are given by

$K_p=\left[\sum_{k=1}^\infty -k^p \log_2\left( 1-\frac{1}{(k+1)^2} \right) \right]^{1/p}$

This is obtained by taking the p-th mean in conjunction with the Gauss-Kuzmin distribution. The value for $K 0$ may be shown to be obtained in the limit of $p\to 0$ .

[edit] 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

K - 1 = 1.74540566240...

[edit] See also

Lévy's constant

[edit] References

David H. Bailey, Jonathan M. Borwein, Richard E. Crandall (1995). "On the Khinchine constant".

Jonathan M. Borwein, David M. Bradley, Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function". J. Comp. App. Math. 121: p.11.