Khinchin's constant is the limiting geometric mean of the partial denominators of the simple continued fraction of almost every real number. Let
![x=[a_0;a_1,...]=a_0+1/(a_1+1/(a_2+1/(a_3+...)))](/images/equations/KhinchinsConstant/NumberedEquation1.svg) |
(1)
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be the simple continued fraction of a "generic" real number 
, where the numbers 
are the partial denominators.
Khinchin (1934) considered the limit of the geometric
mean
 |
(2)
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as 
.
Amazingly, except for a set of measure 0, this limit
is a constant independent of 
given by
 |
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(OEIS A002210), as proved in Kac (1959).
A search for 
-constant primes through decimal prefix length 250000
found no probable primes beyond length 28433 (E. W. Weisstein,
Oct. 17, 2026).
The name is commonly also spelled "Khintchine's constant" (Shanks and Wrench 1959, Bailey et al. 1997).
It is implemented as Khinchin, where its value is cached to 1100-digit precision. However, the numerical value of
is notoriously difficult to calculate to high precision, so computation of more digits
gets increasingly slower.
It is not known if 
is irrational, let alone transcendental.
The equality 
has not been proved for familiar constants such as 
. However, real numbers with
this property can be constructed by explicitly specifying their continued
fractions (Bailey et al. 1997).
The values 
are plotted above for 
to 500 and 
, 
, the Euler-Mascheroni
constant 
,
and the Copeland-Erdős constant 
. Interestingly, the shape of the curves
is almost identical to the corresponding curves for the Lévy
constant.
If 
is the 
th
convergent of the continued
fraction of 
,
then
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(OEIS A086702) for almost all real 
(Lévy 1936, Finch 2003), where
is the natural logarithm of 2. This number
is sometimes called the Lévy constant.
Product expressions for 
include
![K=product_(n=1)^infty[1+1/(n(n+2))]^(lnn/ln2)](/images/equations/KhinchinsConstant/NumberedEquation4.svg) |
(7)
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(Shanks and Wrench 1959; Khinchin 1997, p. 93; Borwein and Bailey 2003, p. 25; Havil 2003, p. 161), where 
is the natural logarithm,
and
 |
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where 
is the forward finite difference operator. The
latter identity follows from three applications of summation
by parts to the natural logarithm of the
usual product definition (W. Gosper, pers. comm., Nov. 14, 2017).
Products such as these can be converted to sums by taking the natural logarithm of both sides and using 
. Sums for 
include
![K=exp[1/(ln2)sum_(m=1)^infty(H_(2m-1)^'[zeta(2m)-1])/m],](/images/equations/KhinchinsConstant/NumberedEquation6.svg) |
(9)
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where 
is the Riemann zeta function and 
is an alternating harmonic
number (Bailey et al. 1997),
![K=exp[1/(ln2)sum_(k=2)^infty((-1)^k(2-2^k)zeta^'(k))/k],](/images/equations/KhinchinsConstant/NumberedEquation7.svg) |
(10)
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where 
is the derivative of the Riemann
zeta function (Gosper, pers. comm., Jun. 25, 1996) and the extremely rapidly
converging sum originally due to Gosper (pers. comm., Jun. 25, 1996) and streamlined
by O. Pavlyk (pers. comm., Apr. 24, 2006) is given by
![K=exp[1/(ln2)(-zeta^'(2,2)+sum_(k=2)^infty2(-1)^kf(k))],](/images/equations/KhinchinsConstant/NumberedEquation8.svg) |
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where
![f(k)=(lnk)/((k+2)k^(k+2))[2^(k+1)_2F_1(1,k+2;k+3;-2/k)-_2F_1(1,k+2;k+3;-1/k)]+((2^k-1)zeta^'(k+1,k))/(k+1),](/images/equations/KhinchinsConstant/NumberedEquation9.svg) |
(12)
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is a Hurwitz zeta function, 
, and 
is a hypergeometric
function.
Khinchin's constant is also given by the integrals
 |  | ![2exp{1/(ln2)int_0^11/(x(1+x))ln[(pix(1-x^2))/(sin(pix))]dx}](/images/equations/KhinchinsConstant/Inline43.svg) |
(13)
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 |  | ![2exp[1/(ln2)int_0^11/(x(1+x))ln[Gamma(2-x)Gamma(2+x)]dx]](/images/equations/KhinchinsConstant/Inline46.svg) |
(14)
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(Shanks and Wrench 1959) and
![K=exp[(pi^2)/(12ln2)+1/2ln2+1/(ln2)int_0^pi(ln(theta|cottheta|)dtheta)/theta].](/images/equations/KhinchinsConstant/NumberedEquation10.svg) |
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Corless (1992) showed that
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with an analogous formula for the Lévy constant.
Real numbers 
for which 
include 
, 
, 
, and the golden ratio 
, plotted above.
The constant 
is the limiting case 
of a class of means defined by
 |
(17)
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for real 
with 
, whose values are given by
![K_p={sum_(k=1)^infty-(k^p)/(ln2)ln[1-1/((k+1)^2)]}^(1/p)](/images/equations/KhinchinsConstant/NumberedEquation13.svg) |
(18)
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(Ryll-Nardzewski 1951, Bailey et al. 1997, Khinchin 1997). An integral representation for 
is given by
 |  | ![[1/(ln2)int_0^1(|_1/t_|^p)/(t+1)dt]^(1/p)](/images/equations/KhinchinsConstant/Inline60.svg) |
(19)
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 |  | ![[1/(ln2)sum_(k=1)^(infty)k^pln(1+1/(k(k+2)))]^(1/p)](/images/equations/KhinchinsConstant/Inline63.svg) |
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for 
,
,
... (Iosifescu and Kraaikamp 2002, p. 231).
The constant
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is sometimes known as the Khinchin harmonic mean and is the 
case of an infinite family of such constants of which 
and 
are the first two members.
Define the following quantity in terms of the 
th partial denominator 
for 
,
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with 
defined by continuity. Then
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for almost all real 
(Khintchine 1934, 1936, Finch 2003). The growth is not bounded
by 
for almost all 
,
since exceptionally large partial denominators
occur infinitely often (Diamond and Vaaler 1986). Furthermore, for 
, the limiting value
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exists and is the constant 
with probability 1 (Rockett
and Szüsz 1992, Khinchin 1997).
