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
(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
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in the continued fraction of a random (or "generic")
real number.
defined for a real number 
by
is the fractional part of 
. This can be defined recursively through
and 
simply the 
th
term of the continued fraction ![x=[a_0;a_1,...]](/images/equations/Gauss-KuzminDistribution/Inline15.svg)
.
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
are illustrated above for 5000 terms of 
, the Euler-Mascheroni
constant 
,
Catalan's constant 
, and the natural logarithm
of 2 
.
, obtaining
.
Using a different method, Lévy (1929) obtained
.
Wirsing (1974) subsequently showed, among other results, that
is a constant known as Gauss-Kuzmin-Wirsing
constant and 
is an analytic function with 
.
![-lg[1-1/((k+1)^2)]](/images/equations/Gauss-KuzminDistribution/Inline34.svg)
![lg[1+1/(k(k+2))]](/images/equations/Gauss-KuzminDistribution/Inline37.svg)
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
![-sum_(k=1)^inftylg[1-1/((k+1)^2)]=1.](/images/equations/Gauss-KuzminDistribution/NumberedEquation8.svg)
