Lévy's constant

In mathematics Lévy's constant (sometimes known as the Khinchin–Lévy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions.^[1] In 1935, the Soviet mathematician Aleksandr Khinchin showed^[2] that the denominators q_n of the convergents of the continued fraction expansions of almost all real numbers satisfy

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

for some constant γ. Soon afterward, in 1936, the French mathematician Paul Lévy found^[3] the explicit expression for the constant, namely

$\gamma =e^{{\pi ^{2}/(12\ln 2)}}=3.275822918721811159787681882\ldots .$

The term "Lévy's constant" is sometimes used to refer to $\pi ^{2}/(12\ln 2)$ (the logarithm of the above expression), which is approximately equal to 1.1865691104….

The base-10 logarithm of Lévy's constant, which is approximately 0.51532041…, is half of the reciprocal of the limit in Lochs' theorem.

See also

• Khinchin's constant

References

1. A. Ya. Khinchin; Herbert Eagle (transl.) (1997), Continued fractions, Courier Dover Publications, p. 66, ISBN 978-0-486-69630-0
2. [Reference given in Dover book] "Zur metrischen Kettenbruchtheorie," Compositio Matlzematica, 3, No.2, 275–285 (1936).
3. [Reference given in Dover book] P. Levy, Théorie de l'addition des variables aléatoires, Paris, 1937, p. 320.

Further reading

• Khinchin, A. Ya. Continued Fractions. Dover. ISBN 0-486-69630-8.

External links

• Weisstein, Eric W. "Khinchin–Lévy Constant". MathWorld.

• Decimal expansion of Levy's constant: (sequence A086702 in the OEIS)