## Gauss-Kuzmin-Wirsing Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let be a random number from written as a simple Continued Fraction

 (1)

Define
 (2)

Gauß (1800) showed that if is the probability that , then
 (3)

Kuzmin (1928) published the first proof, which was subsequently improved by Lévy (1929). Wirsing (1974) showed, among other results, that
 (4)

where and is an analytic function with This constant is connected to the efficiency of the Euclidean Algorithm (Knuth 1981).

References

Babenko, K. I. On a Problem of Gauss.'' Soviet Math. Dokl. 19, 136-140, 1978.

Daudé, H.; Flajolet, P.; and Vallée, B. An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction.'' Submitted.

Durner, A. On a Theorem of Gauss-Kuzmin-Lévy.'' Arch. Math. 58, 251-256, 1992.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/kuzmin/kuzmin.html

Flajolet, P. and Vallée, B. On the Gauss-Kuzmin-Wirsing Constant.'' Unpublished memo. 1995. http://pauillac.inria.fr/algo/flajolet/Publications/gauss-kuzmin.ps.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd ed. Reading, MA: Addison-Wesley, 1981.

MacLeod, A. J. High-Accuracy Numerical Values of the Gauss-Kuzmin Continued Fraction Problem.'' Computers Math. Appl. 26, 37-44, 1993.

Wirsing, E. On the Theorem of Gauss-Kuzmin-Lévy and a Frobenius-Type Theorem for Function Spaces.'' Acta Arith. 24, 507-528, 1974.