Golomb–Dickman constant

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

${\displaystyle \lambda =0.62432998854355087099293638310083724\dots }$ (sequence A084945 in the OEIS)

It is not known whether this constant is rational or irrational.[1]

Definitions

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

${\displaystyle \lambda =\lim _{n\to \infty }{\frac {a_{n}}{n}}.}$

In the language of probability theory, ${\displaystyle \lambda n}$  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,

${\displaystyle \lambda =\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=2}^{n}{\frac {\log(P_{1}(k))}{\log(k)}},}$

where ${\displaystyle P_{1}(k)}$  is the largest prime factor of k. So if k is a d digit integer, then ${\displaystyle \lambda d}$  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 ${\displaystyle \lambda }$ . More precisely,

${\displaystyle \lambda =\lim _{n\to \infty }{\text{Prob}}\left\{P_{2}(n)\leq {\sqrt {P_{1}(n)}}\right\}}$

where ${\displaystyle P_{2}(n)}$  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: XX to any element x of this set, it eventually enters a cycle, meaning that for some k we have ${\displaystyle f^{n+k}(x)=f^{n}(x)}$  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[2] proved that

${\displaystyle \lim _{n\to \infty }{\frac {b_{n}}{\sqrt {n}}}={\sqrt {\frac {\pi }{2}}}\lambda .}$

Formulae

There are several expressions for ${\displaystyle \lambda }$ . These include:

${\displaystyle \lambda =\int _{0}^{1}e^{\mathrm {Li} (t)}\,dt}$

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

${\displaystyle \lambda =\int _{0}^{\infty }e^{-t-E_{1}(t)}\,dt}$

where ${\displaystyle E_{1}(t)}$  is the exponential integral, and

${\displaystyle \lambda =\int _{0}^{\infty }{\frac {\rho (t)}{t+2}}\,dt}$

and

${\displaystyle \lambda =\int _{0}^{\infty }{\frac {\rho (t)}{(t+1)^{2}}}\,dt}$

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