Probabilistic interpretation for representation of unity using the zeta function

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.