Vallée Constant

The Vallée constant is the limit, as , of the nth roots of the probabilities that two independent random variables and , each uniformly distributed on , have the same first partial denominators in their regular continued fraction representations. Let be the first index at which their partial denominators differ. The mean number of iterations needed to determine whether or is given by

			(1)
			(2)
			(3)
			(4)

(OEIS A074903; Finch 2003, p. 161), where is a polylogarithm and is the Riemann zeta function.

Let for . The first few values are

			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

The value

(13)

(OEIS A143302) is then known as the Vallée constant (Finch 2003, p. 161).