Zeta distributions, why is it a probability measure and why are these variables independent? I dont understand why are the Zeta distributions family:
$$P(X=n):=\frac{1}{\zeta(s)n^s}$$ 
 are probability measures (which is additive) on all of the partial subsets of N?
and why are the variables $\chi_p(n)=\left\{\begin{matrix}
1 & p|n \\ 
0 & \text{esle}
\end{matrix}\right.$ independent? 
Thank you for your help...
 A: Every family $(a_n)_{n\in\mathbb{N}}$ of non-negative real numbers with sum $1$ defines a probability measure on the $\sigma$-algebra of all subsets of $\mathbb{N}$ per
$$P(A) = \sum_{n \in A} a_n.$$
Since by definition of $\zeta(s)$ we have
$$\sum_{n=1}^\infty \frac{1}{\zeta(s)n^s} = 1,$$
and you get a probability measure from these weights.
To see that the variables $\chi_p$ - where $p$ is a prime - are independent, compute
$$\sum_{m \mid n} \frac{1}{\zeta(s)n^s} = \sum_{k=1}^\infty \frac{1}{\zeta(s)(k\cdot m)^s} = \frac{1}{m^s}\sum_{k=1}^\infty \frac{1}{\zeta(s)k^s} = \frac{1}{m^s},$$
and note that $n$ is a multiple of the two primes $p$ and $q$ if and only if it is a multiple of $p\cdot q$, whence $E[\chi_p \cdot \chi_q] = \frac{1}{(pq)^s} = E[\chi_p]\cdot E[\chi_q]$. That generalises to all finite families of distinct primes, $n$ is a multiple of all primes $p_1,\, p_2,\, \dotsc,\, p_k$ if and only if $n$ is a multiple of the product $p_1 \cdot p_2 \cdot \dotsc \cdot p_k$, and hence
$$E\left[\prod_{j=1}^k \chi_{p_j}\right] = \frac{1}{\prod\limits_{j=1}^k p_j^s} = \prod_{j=1}^k E[\chi_{p_j}],$$
which means the family $\{\chi_{p_j} : 1 \leqslant j \leqslant k\}$ is independent. Since all finite subfamilies are independent, the entire family $\{ \chi_p : p \text{ prime}\}$ is independent.
Note: We could more generally define
$$\tilde{\chi}_m(n) = \begin{cases} 1 &, m \mid n\\ 0 &, m \nmid n\end{cases}$$
and find that $\tilde{\chi}_m$ and $\tilde{\chi}_k$ are independent if and only if $m$ and $k$ are coprime, since $\tilde{\chi}_m\cdot \tilde{\chi}_k = \tilde{\chi}_{\operatorname{lcm}(m,k)}$. For every family $F$ of pairwise coprime numbers, the family $\{\tilde{\chi}_m : m \in F\}$ is then independent, which can be seen as above.
