# 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

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

The term "Lévy's constant" is sometimes used to refer to (the logarithm of the above expression), which is approximately equal to 1.1865691104… The value derives from the asymptotic expectation of the logarithm of the ratio of successive denominators, using the Gauss-Kuzmin distribution. In particular, the ratio has the asymptotic density function

for and zero otherwise. This gives Lévy's constant as

.

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[edit]

## References[edit]

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

## Further reading[edit]

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

## External links[edit]

- Weisstein, Eric W. "Lévy Constant".
*MathWorld*. - OEIS sequence A086702 (Decimal expansion of Lévy's constant)

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