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. In 1935, the Soviet mathematician Aleksandr Khinchin showed 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 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

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)