The Gauss-Kuzmin distribution is the limiting distribution of occurrences of a positive integer 
in the continued fraction
of a random real number.
For an irrational number ![y=[a_0;a_1,...]](/images/equations/Gauss-KuzminDistribution/Inline2.svg)
, define the complete quotients 
for 
by
 |  |  |
(1)
|
 |  |  |
(2)
|
so 
is the fractional part of 
. This can be defined recursively through
 |
(3)
|
and
 |
(4)
|
with 
,
where 
is the floor function and 
is the 
th term of the continued
fraction of 
.
The distribution of 
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 
with the uniform distribution on (0, 1),
then
 |
(5)
|
for 
,
where 
(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
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(7)
|
with 
.
Using a different method, Lévy (1929) obtained
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(8)
|
with 
.
Wirsing (1974) subsequently showed, among other results, that
 |
(9)
|
where 
is a constant known as the Gauss-Kuzmin-Wirsing
constant and 
is an analytic function
with 
.
It follows from Gauss's result that
 |  | ![-lg[1-1/((k+1)^2)]](/images/equations/Gauss-KuzminDistribution/Inline38.svg) |
(10)
|
 |  | ![lg[1+1/(k(k+2))]](/images/equations/Gauss-KuzminDistribution/Inline41.svg) |
(11)
|
for positive integer 
(Bailey et al. 1997; Havil 2003, p. 158). The
name "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) |
(12)
|
