(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,
Corless, R. M. "Continued Fractions and Chaos." Amer. Math. Monthly99, 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.
Monthly46, 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.