The Riemann zeta function and Haar measure on the profinite integers

Question

In an answer to a question on MU about the Riemann zeta function, I sketched a proof that the probability distribution on $\mathbb{N}$ which assigns $n$ the probability

$$\frac{ \frac{1}{n^s} }{\zeta(s)}$$

(henceforth called the zeta distribution with parameter $s$) where $s > 1$ is the unique family of probability distributions on $\mathbb{N}$ satisfying the following three requirements:

• The exponent of $p$ and the exponent of $q$ in the prime factorization of $n$ are chosen independently for all pairs of primes $p \neq q$.
• The exponent of a particular prime $p$ is geometrically distributed.
• The probabilities are monotonically decreasing as a function of $n$.

The basic motivation for the first requirement is the Chinese Remainder Theorem. I can think of two motivations for the second requirement: first, that geometric distributions are the maximally entropic distributions on $\mathbb{N}_{\ge 0}$ with a given mean, and second, that (if I'm not mistaken) one naturally gets a geometric distribution from Haar measure on the $p$-adic integers.

In fact, the distribution one gets from Haar measure on the $p$-adic integers is the one in which a $p$-adic integer is divisible by exactly $p^k$ with probability $(1 - p^{-1}) p^{-k}$. This is essentially the $s \to 1$ limit of the zeta distribution above, which I gave as a reason one might deduce that this limit is important from first principles.

It seems like it should be possible to combine the motivations for the first two requirements into a statement about Haar measure on the profinite integers $\prod_p \mathbb{Z}_p$, except that I don't know exactly what kind of statement I'd be looking for, so I thought I'd ask here.

Question: Complete the following statement. It is natural to study the $s \to 1$ limit of the zeta distribution because (some statement about Haar measure on the profinite integers, maybe with "Tate's thesis" thrown in somewhere).

Answer 1

...because for sets with logarithmic density, the logarithmic density and the $s\to1$ zeta-measure agree. And for sets with natural density, the logarithmic and natural densities are the same.

Answer 2

"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, 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.