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
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]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/LevyConstant/Inline39.gif)
." §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.
Weisstein, Eric W. "Lévy Constant." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LevyConstant.html
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