Lévy constant

After Paul Lévy called Lévy constant or Lévysche number is a mathematical constant that is at the limit formation of continued fractions play a role: Considering the th root of th denominator of the continued fraction expansion of a real number , so there are in almost all a limit when going towards infinity: ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle q_{n}}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle n}$

${\displaystyle \lim _{n\to \infty }{q_{n}}^{1/n}=\gamma }$

This was shown in 1935 by the Soviet mathematician Aleksandr Khinchin . In the following year, the French mathematician Paul Lévy found an explicit representation for Lévy's constant, namely:

${\displaystyle \gamma =e^{\pi ^{2}/(12\ln 2)}=3{,}275822918721811159787681882\ldots }$

The expression in it

${\displaystyle \beta ={\frac {\pi ^{2}}{12\ln 2}}=1{,}1865691104\ldots }$

was called the Khinchin-Lévy constant , although the names are not used uniformly.

The double logarithm of Lévy's constant is equal to the limit that occurs in the set of holes for the decimal system.

RM Corless showed

${\displaystyle \beta ={\frac {1}{2}}\int _{0}^{1}{\frac {\ln x^{-1}}{(x+1)\ln 2}}\mathrm {d} x}$

and put the Lévy constant in connection with the Khinchin constant .

Web links

• Eric W. Weisstein : Khinchin – Lévy Constant . In: MathWorld (English).
• Follow A086702 in OEIS

Individual evidence

1. ↑ Aleksandr Khinchin : On the metric continued fraction theory . In: Compositio Mathematica , 3, 1936, No. 2, pp. 275-285. uni-goettingen.de ( Memento of the original from May 25, 2015 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
2. ↑ P. Lévy: Sur le développement en fraction continue d'un nombre choisi au hasard . In: Compositio Mathematica , 1936, pp. 286-303. Reprinted in Œuvres de Paul Lévy , Vol. 6. Gauthier-Villars, Paris 1980, pp. 285-302.
3. ^ RM Corless: Continued Fractions and Chaos . In: American Mathematical Monthly , Number 99, 1992, pp. 203-215.