Golomb–Dickman constant

In mathematics, the Golomb–Dickman constant, named after Solomon W. Golomb and Karl Dickman, arises in the theory of random permutations and in number theory. Its value is


 * $$\lambda = 0.62432 99885 43550 87099 29363 83100 83724\dots$$

It is not known whether this constant is rational or irrational.

Definitions
Let an be the average &mdash; taken over all permutations of a set of size n &mdash; of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is


 * $$\lambda = \lim_{n\to\infty} \frac{a_n}{n}.$$

In the language of probability theory, $$\lambda n$$ is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size n.

In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer. More precisely,
 * $$\lambda = \lim_{n\to\infty} \frac1n \sum_{k=2}^n \frac{\log(P_1(k))}{\log(k)},$$

where $$P_1(k)$$ is the largest prime factor of k. So if k is a d digit integer, then $$\lambda d$$ is the asymptotic average number of digits of the largest prime factor of k.

The Golomb–Dickman constant appears in number theory in a different way. What is the probability that second largest prime factor of n is smaller than the square root of the largest prime factor of n? Asymptotically, this probability is $$\lambda$$. More precisely,
 * $$\lambda = \lim_{n\to\infty} \text{Prob}\left\{P_2(n) \le \sqrt{P_1(n)}\right\}$$

where $$P_2(n)$$ is the second largest prime factor 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 &rarr; X to any element x of this set, it eventually enters a cycle, meaning that for some k we have $$f^{n+k}(x) = f^n(x)$$ for sufficiently large n; the smallest k with this property is the length of the cycle. Let bn be the average, taken over all functions from a set of size n to itself, of the length of the largest cycle. Then Purdom and Williams proved that


 * $$  \lim_{n\to\infty} \frac{b_n}{\sqrt{n}} = \sqrt{\frac{\pi}{2} } \lambda. $$

Formulae
There are several expressions for $$\lambda$$. These include:


 * $$\lambda = \int_0^1 e^{\mathrm{Li}(t)} \, dt $$

where $$\mathrm{Li}(t)$$ is the logarithmic integral,


 * $$\lambda = \int_0^\infty e^{-t - E_1(t)} \, dt $$

where $$E_1(t)$$ is the exponential integral, and


 * $$\lambda = \int_0^\infty \frac{\rho(t)}{t+2} \, dt $$

and


 * $$\lambda = \int_0^\infty \frac{\rho(t)}{(t+1)^2} \, dt $$

where $$\rho(t)$$ is the Dickman function.