I have some difficulty starting with this question:
For $s>1$, set $\zeta(s)=\sum_{n=1}^\infty n^{-s}$, and define $p: \mathbb{N} \rightarrow [0,1]$ by $p(n)=\frac{n^{-s}}{\zeta(s)}$. One can define a probability space with $\Omega=\mathbb{N}$, $\mathscr{F}$ the $\sigma$-algebra of all subsets of $\mathbb{N}$, with $\mathscr{P}[A]=\sum_{j\in A} p(j)$.
(a) let $A_k$ be the event that a randomly chosen number is divisible by $k$. Show that $$\mathscr{P}[A_k]=\frac{1}{k^s}.$$ (b) Let $S$ be the event that a randomly chosen number is a square. Show that $$\mathscr{P}[S]=\frac{\zeta(2s)}{\zeta(s)}.$$
For a) I had some ideas of taking sets such as {$k,2k,3k,\ldots nk$} but I somehow don't see how to go about this problem in general. Any help would be great!
Hint: "randomly chosen" means randomly chosen with probabilities given by $\mathscr P$. So ${\mathscr P}\{ k, 2k, 3k \ldots\} = \sum_{j=1}^\infty {\mathscr P}\{ j k\} = \ldots$ and ${\mathscr P}\{1^2, 2^2, 3^2, \ldots\} = \sum_{j=1}^\infty {\mathscr P}\{j^2\} = \ldots$