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"This is also the unique nontrivial distribution on the positive integers that makes the events "p divides X" and "q divides X" independent whenever p,q are distinct primes."

Kevin, consider $X = \prod_{i=1}^\infty p_i^{N_i}$ where the $N$'s are independent, and $N_i$ has a Poisson distribution with parameter $p_i^{-s}$. It turns out this distribution is related to the exponential of the prime zeta function.

In fact, whenever you construct a probability distribution by normalising the terms of a convergent Dirichlet series with positive, pultiplicitive multiplicitive coefficients, this property shows up. It's the subject of my MPhil thesis, which I may manage to post up on the arxiv sometime this century.

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"This is also the unique nontrivial distribution on the positive integers that makes the events "p divides X" and "q divides X" independent whenever p,q are distinct primes."

Kevin, consider $X = \prod_{i=1}^\infty p_i^{N_i}$ where the $N$'s are independent, and $N_i$ has a Poisson distribution with parameter $p_i^{-s}$. It turns out this distribution is related to the exponential of the prime zeta function.

In fact, whenever you construct a probability distribution by normalising the terms of a convergent Dirichlet series with positive, pultiplicitive coefficients, this property shows up. It's the subject of my MPhil thesis, which I may manage to post up on the arxiv sometime this century.