Lévy Constant

The Lévy constant is the limiting value of the nth root of the denominator of the th convergent of the simple continued fraction of a real number ,

			(1)
			(2)
			(3)

(OEIS A086702) for all outside a set 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.