In mathematics, the Golomb–Dickman constant, named after Solomon W. Golomb and Karl Dickman, is a mathematical constant, which arises in the theory of random permutations and in number theory. Its value is
λ=0.62432998854355087099293638310083724...
It is not known whether this constant is rational or irrational.[1]
It's simple continued fraction is given by
[0;1,1,1,1,1,22,1,2,3,1,...]
Let an be the average - taken over all permutations of a set of size n - of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is
λ=\limn\toinfty
| an | |
| n |
.
In the language of probability theory,
λn
In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer. More precisely,
λ=\limn\toinfty
| 1n | |
| \sum |
| n | |
| k=2 |
| log(P1(k)) | |
| log(k) |
,
P1(k)
λd
The Golomb–Dickman constant appears in number theory in a different way. What is theprobability that second largest prime factor of n is smaller than the square root of the largest prime factor of n? Asymptotically, this probability is
λ
λ=\limn\toinftyProb\left\{P2(n)\le\sqrt{P1(n)}\right\}
P2(n)
The Golomb-Dickman constant also arises when we consider the average length of the largest cycle of any function from a finite set to itself. If X is a finite set, if we repeatedly apply a function f: X → X to any element x of this set, it eventually enters a cycle, meaning that for some k we have
fn+k(x)=fn(x)
\limn\toinfty
| bn | |
| \sqrt{n |
There are several expressions for
λ
λ=
| 1 | |
| \int | |
| 0 |
eli(t)dt
where
li(t)
λ=
| infty | |
| \int | |
| 0 |
| -t-E1(t) | |
| e |
dt
where
E1(t)
λ=
| infty | |
| \int | |
| 0 |
| \rho(t) | |
| t+2 |
dt
and
λ=
| infty | |
| \int | |
| 0 |
| \rho(t) | |
| (t+1)2 |
dt
where
\rho(t)