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Lévy Constant


The nth root of the denominator B_n of the nth convergent A_n/B_n of a number x tends to a constant

lim_(n->infty)B_n^(1/n)=e^beta
(1)
=e^(pi^2/(12ln2))
(2)
=3.275823...
(3)

(OEIS A086702) for all but a set of x of measure zero (Lévy 1936, Lehmer 1939), where

beta=(pi^2)/(12ln2)
(4)
=1.1865691104...
(5)

Some care is needed in terminology and notation related to this constant. Most authors call e^beta "Lévy's constant" (e.g., Le Lionnais 1983, p. 51; Sloane) and some (S. Plouffe) call beta the "Khinchin-Lévy constant." Other authors refer to e^beta (e.g., Finch 2003, p. 60) or beta (e.g., Wu 2008) without specifically naming the expression in question.

Taking the multiplicative inverse of beta gives another related constant,

beta^(-1)=(12ln2)/(pi^2)
(6)
=0.8427659...
(7)

(OEIS A089729).

Corless (1992) showed that

 beta=int_0^1(lnx^(-1))/((x+1)ln2)dx,
(8)

with an analogous formula for Khinchin's constant.

The Lévy Constant e^beta is related to Lochs' constant L by

 beta=(ln10)/(2L)
(9)

or

 e^beta=10^(1/(2L)).
(10)
Khinchin-LevyConstant

The plot above shows B_n^(1/n) for the first 500 terms in the continued fractions of pi, sin1, the Euler-Mascheroni constant gamma, and the Copeland-Erdős constant C. Interestingly, the shape of the curves is almost identical to the corresponding curves for Khinchin's constant


See also

Continued Fraction, Convergent, Gauss-Kuzmin-Wirsing Constant, Khinchin's Constant, Lochs' Constant, Lochs' Theorem

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References

Corless, R. M. "Continued Fractions and Chaos." Amer. Math. Monthly 99, 203-215, 1992.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 60 and 156, 2003.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 51, 1983.Lehmer, D. H. "Note on an Absolute Constant of Khintchine." Amer. Math. Monthly 46, 148-152, 1939.Lévy, P. "Sur le développement en fraction continue d'un nombre choisi au hasard." Compositio Math. 3, 286-303, 1936. Reprinted in Œuvres de Paul Lévy, Vol. 6. Paris: Gauthier-Villars, pp. 285-302, 1980.Rockett, A. M. and Szüsz, P. "The Khintchine-Lévy Theorem for RadicalBox[{B, _, n}, n]." §5.9 in Continued Fractions. New York: World Scientific, pp. 163-166, 1992.Sloane, N. J. A. Sequences A086702 and A089729 in "The On-Line Encyclopedia of Integer Sequences."Wu. J. "An Iterated Logarithm Law Related to Decimal and Continued Fraction Expansions." Monatsh. f. Math. 153, 83-87, 2008.

Cite this as:

Weisstein, Eric W. "Lévy Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LevyConstant.html

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