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Khinchin-Lévy Constant

download Mathematica trial version Khinchin-LevyConstant.nb

The nth radical root of the denominator of the nth convergent of a number x tends to a constant


(Sloane's A086702) for all but a set of x of measure zero (Lévy 1936, Lehmer 1939). Plouffe called the exponent the Khinchin-Lévy constant. Taking its multiplicative inverse gives (Sloane's A089729).

The plot above shows for the first 500 terms in the continued fractions of , , the Euler-Mascheroni constant , and the Copeland-Erdos constant C. Interestingly, the shape of the curves is almost identical to the corresponding curves for Khinchin's constant

Gauss-Kuzmin-Wirsing Constant, Khinchin's Constant

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References

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 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 uvres de Paul Lévy, Vol. 6. Paris: Gauthier-Villars, pp. 285-302, 1980.

Rockett, A. M. and Szüsz, P. "The Khintchine-Lévy Theorem for ." §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." http://www.research.att.com/~njas/sequences/.




cite this as

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



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