Khintchine's Constant

Let x be a real number. Expand x (uniquely) as a regular continued fraction:

where is an integer and are positive integers.

What is the average behavior of , where k>0 is arbitrary? Consider, for example, the geometric mean

in the limit as n increases without bound. Khintchine proved that

a constant, for almost all real numbers x. This means that the set of exceptions x to Khintchine's result is of Lebesgue measure zero.

The infinite product representation of K converges very slowly. Lehmer and Shanks & Wrench discussed numerical evaluation procedures for K, culminating in a 155 decimal place calculation by Wrench. More recently, Gosper computed 1111 places and Bailey, Borwein & Crandall surpassed even this with 7350 places.

Related ideas include the asymptotic behavior of the relatively prime positive integers and given by

Lévy determined that

for almost all real x.

A closely related topic is the Gauss-Kuzmin-Wirsing constant.

Plouffe gave a highly accurate approximation of K in the Inverse Symbolic Calculator web pages, as well as an approximation of Lévy's constant.

Gosper additionally wrote, "In the early 70's, Gene Salamin derived the variance of this distribution, which involved , but he thought the matter unworthy of publication." Here is more information.

Philippe Flajolet has kindly visited and offers some comments.

More details and references (contact Steven Finch).
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