Gauss-Kuzmin Distribution

The Gauss-Kuzmin distribution is the distribution of occurrences of a positive integer  in the continued fraction of a random (or "generic") real number.

Consider  defined for a real number  by

			(1)
			(2)

so  is the fractional part of . This can be defined recursively through

(3)

and

(4)

with  and  simply the th term of the continued fraction .

The distribution  was considered by Gauss in a letter to Laplace dated January 30, 1812. In this letter, Gauss said that he could prove by a simple argument that if , sometimes denoted  (Havil 2003, p. 156), is the probability that  for a random , then

(5)

(Rockett and Szüsz 1992, pp. 151-152; Knuth 1998, p. 341; Havil 2003, p. 157). Histograms of  are illustrated above for 5000 terms of , the Euler-Mascheroni constant , Catalan's constant , and the natural logarithm of 2 .

However, Gauss was unable to describe the behavior of the correction term in

(6)

Kuz'min (1928) published the first analysis of the asymptotic behavior of , obtaining

(7)

with . Using a different method, Lévy (1929) obtained

(8)

with . Wirsing (1974) subsequently showed, among other results, that

(9)

where  is a constant known as Gauss-Kuzmin-Wirsing constant and  is an analytic function with .

It follows from Gauss's result that

			(10)
			(11)

(Bailey et al. 1997; Havil 2003, p. 158), where  and "Kuzmin" is sometimes also written as "Kuz'min." The plot above shows the distribution of the first 500 terms in the continued fractions of , , the Euler-Mascheroni constant , and the Copeland-Erdős constant . The distribution is properly normalized, since

(12)

REFERENCES:

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

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "On the Khintchine Constant." Math. Comput. 66, 417-431, 1997.

Daudé, H.; Flajolet, P.; and Vallé, B. "An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction." Combin. Probab. Comput. 6, 397-433, 1997.

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

Finch, S. R. "Gauss-Kuzmin-Wirsing Constant." §2.17 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 151-156, 2003.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 155-159, 2003.

Khinchin, A. Ya. "Gauss's Problem and Kuz'min's Theorem." §15 in Continued Fractions. New York: Dover, pp. 71-86, 1997.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998.

Kuz'min, R. O. "A Problem of Gauss." Dokl. Akad. Nauk, Ser. A, 375-380, 1928.

Kuz'min, R. O. "Sur un problème de Gauss." Anni Congr. Intern. Bologne 6, 83-89, 1928.

Lévy, P. "Sur les lois de probabilité dont dependent les quotients complets et incomplets d'une fraction continue." Bull. Soc. Math. France 57, 178-194, 1929.

Lévy, P. "Sur les lois de probabilité dont dependent les quotients complets et incomplets d'une fraction continue." Bull. Soc. Math. France 57, 178-194, 1929.

Rockett, A. M. and Szüsz, P. "The Gauss-Kuzmin Theorem." §5.5 in Continued Fractions. New York: World Scientific, pp. 151-155, 1992.

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