Lévy Constant

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

			(1)
			(2)
			(3)

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

			(4)
			(5)

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

Taking the multiplicative inverse of  gives another related constant,

			(6)
			(7)

(OEIS A089729).

Corless (1992) showed that

(8)

with an analogous formula for Khinchin's constant.

The Lévy Constant  is related to Lochs' constant  by

(9)

or

(10)

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

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 ." §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.