Lévy's constant
Encyclopedia
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 fraction
s.
In 1935, the Soviet mathematician
Aleksandr Khinchin showed that the denominators qn 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 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 base-10 logarithm of Lévy's constant, which is approximately 0.51532941…, is half of the reciprocal of the limit in Lochs' theorem
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
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 fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s.
In 1935, the Soviet mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Aleksandr Khinchin showed that the denominators qn of the convergents of the continued fraction expansions of almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
real numbers satisfy
for some constant γ. Soon afterward, in 1936, the French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician Paul Lévy
Paul Pierre Lévy
Paul Pierre Lévy was a Jewish French mathematician who was active especially in probability theory, introducing martingales and Lévy flights...
found 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 base-10 logarithm of Lévy's constant, which is approximately 0.51532941…, is half of the reciprocal of the limit in Lochs' theorem
Lochs' theorem
In number theory, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction expansion of a typical real number. The theorem was proved by Gustav Lochs in 1964....
.
External links
- Decimal expansion of Levy's constant: