3
$\begingroup$

I was thinking about the fact that $\frac{1}{\zeta(n)}$ is the probability that $n$ randomly chosen integers are coprime, and wondered if there were “natural” ways in which $e^{-n}$ (for n>0) and $\frac{1}{\Gamma(n)} $(provided that $0 < \frac{1}{\Gamma(n)} < 1$) might be interpreted as probabilities with respect to a property of some structure on $n$ objects?

  • 2
    If $n$ is a positive non-zero integer, $\Gamma(n)=(n-1)!$. I guess you can think of some probabilistic interpretation for that.2011-03-01

1 Answers 1

3

This isn't exactly what you asked for, but with $H_n = \sum_{k=1}^n \frac{1}{k}$ the $n^{th}$ harmonic number, it turns out that $e^{-H_n}$ is asymptotically the probability that a permutation of a large set has no cycles of length at most $n$. This generalizes the well-known asymptotic for derangements when $n = 1$. I think you can prove it by inclusion-exclusion, but there is a much cleaner proof using the exponential formula.