Gauss-Kuzmin-Wirsing Constant

Wirsing (1974) showed, among other results, that if F_n(x) is the Gauss-Kuzmin distribution, then


where lambda=0.3036630029... (OEIS A038517; Knuth 1998, p. 350) and Psi(x) is an analytic function with Psi(0)=Psi(1)=0.

lambda was computed to about 30 decimal places by Flajolet and Vallée (1995) and to 100 places by Sebah (unpublished). Briggs (2003) computed lambda as the negative of the second largest (in absolute value) eigenvalue of the (n+1)×(n+1) matrix defined by

 M_(jk)=((-1)^j)/(j!(-2)^k)sum_(i=0)^k(k; i)(-2)^i(i+2)_j[zeta(i+j+2)(2^(i+j+2)-1)-2^(i+j+2)]

for 0<=j,k<=n, where (k; i) is a binomial coefficient, (x)_n is a Pochhammer symbol, and zeta(z) is the Riemann zeta function. For example,

 M_2=[1/2(pi^2-8) 7zeta(3)-1/4pi^2-6; 16-14zeta(3) 7zeta(3)-1/2pi^4+40].

Briggs (2003) used n=800 and a precision of 1300 bits to obtain 385 digits.

This constant is connected to the efficiency of the Euclidean algorithm. It has continued fraction [0, 3, 3, 2, 2, 3, 13, 1, 174, ...] (OEIS A007515; Knuth 1998, p. 350).

See also

Continued Fraction, Euclidean Algorithm, Gauss-Kuzmin Distribution, Khinchin's Constant, Lévy Constant

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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.Briggs, K. "A Precise Computation of the Gauss-Kuzmin-Wirsing Constant." Preliminary report. 2003 July 8.é, H.; Flajolet, P.; and Vallé, B. "An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction." Combin. Probab. Comput. 6, 397-433, 1997.Finch, S. R. "Gauss-Kuzmin-Wirsing Constant." §2.17 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 151-156, 2003.Flajolet, P. and Vallée, B. "On the Gauss-Kuzmin-Wirsing Constant." Unpublished memo. 1995., D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, p. 341, 1998.MacLeod, A. J. "High-Accuracy Numerical Values of the Gauss-Kuzmin Continued Fraction Problem." Computers Math. Appl. 26, 37-44, 1993.Mayer, D. H. "Continued Fractions and Related Transformations." In Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces. Papers from the Workshop on Hyperbolic Geometry and Ergodic Theory held in Trieste, April 17-28, 1989 (Ed. T. Bedford, M. Keane, and C. Series). New York: Clarendon Press, pp. 175-222, 1991.Plouffe, S. "The Gauss-Kuzmin-Wirsing Constant.", N. J. A. Sequences A007515/M2267 and A038517 in "The On-Line Encyclopedia of Integer Sequences."Wirsing, E. "On the Theorem of Gauss-Kuzmin-Lévy and a Frobenius-Type Theorem for Function Spaces." Acta Arith. 24, 507-528, 1974.

Referenced on Wolfram|Alpha

Gauss-Kuzmin-Wirsing Constant

Cite this as:

Weisstein, Eric W. "Gauss-Kuzmin-Wirsing Constant." From MathWorld--A Wolfram Web Resource.

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