Probabilistic interpretation for representation of unity using the zeta function

Question

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4):

$$1=\sum_{n=2}^\infty (\zeta(n)-1).$$

There are some generalizations in the linked paper as well.

Question: Is there an interesting probabilistic interpretation of this that comes up somewhere? In other words, a random variable $X$ such that $P(X=n)=\zeta(n)-1$, for $n\geq 2$.

Note that $\zeta(n)>1$ for all $n>1$, and $0<\zeta(n)-1<1$ for all $n\geq 2$.

Answer 1

There is a family of distributions called Zipf (sometimes zeta) distributions. A random variable $X$ with this distribution satisfies $$\mathbb P(X=n)=\frac{n^{-s}}{\zeta(s)},\qquad\qquad n=1,2,3,\dots$$ where $s>1$ is a parameter of the distribution. A main interesting property of this is that the prime factors of $X$ are independent, that is the events $[p_1|x],[p_2|X],\dots,[p_k|X]$ are independent for primes $p_1,\dots,p_k.$

I don't know about the case your talking about although it seems to be a special case of the distribution discussed in the paper A Probability Distribution Associated with the Hurwitz Zeta Function from Proceedings of the American Mathematical Society, vol. 99 no. 4 (Apr. 1987), p. 757-759. You may be able to access it on JSTOR.

EDIT: Here it is from the AMS.