### The Golomb–Dickman Constant

#### Posted by John Baez

I’m falling in love with random permutations. There’s something both simple and fairly deep about them. I like to visualize a random permutation as a “gas of cycles”, governed by the laws of statistical mechanics. I haven’t gotten very with this analogy yet.

Here’s one of the many cute facts about random permutations. Let $a_n$ be the average length of the longest cycle in a permutation, averaged over all permutations of an $n$-element set. Then $a_n$ is asymptotically equal to $\lambda n$ where

$\lambda \approx 0.6243299885 \dots$

is called the Golomb–Dickman constant. There’s a cool formula for this constant: $\lambda = \int_0^1 e^{\mathrm{Li}(x)} dx \; \; where \; \; Li(x) = \int_0^x \frac{dt}{\ln t}$ But nobody has been able to prove that it’s irrational!

The Golomb–Dickman constant also shows up in number theory… in a very similar way! If you randomly choose a huge $n$-digit integer, the average number of digits of its largest prime factor is asymptotic to $\lambda n$.

So, there’s a connection between prime factorizations and random permutations! And this fact is both simpler and more interesting to me than the Golomb–Dickman constant. You can read more about this in Sections 3.10-3.12 here:

- Jeffrey Lagarias, Euler’s constant: Euler’s work and modern developments.

The Golomb–Dickman constant is a kind of relative of Euler’s constant, though there’s no known formula expressing one in terms of the other.

Here’s another appearance of this constant. Say you randomly choose a function from a huge $n$-element set to itself. Then the average length of its longest periodic orbit is asymptotic to

$\lambda \; \sqrt{\frac{\pi n}{2}}$