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A somewhat surprising occurence, which can be seen immediatelly via the Euler product, appears in the study of visible points of lattice.

Given a lattice $\Gamma \subset \mathbb R^d$, meaning $\Gamma=\mathbb Z v_1 \oplus ... \oplus\mathbb Z v_d$ for some $\mathbb R$ basis $v_1,.., v_d$ of $\mathbb R^d$, the visible points of $\Gamma$ are defined as $$V:= \{ z=n_1v_1+...+n_dv_d : n_1,.., n_d \in \mathbb Z , \mbox{ gcd } (n_1,.., n_d)=1 \}$$

The, we have the following result, (see Prop.~6 in Diffraction from visible lattice points and k-th power free integers)

Proposition The natural density of $V$ is $$ \mbox{dens}(V)=\frac{1}{ \det(A) \zeta(d) } $$ where $A$ is the matrix with columns $v_1,v_2,...,v_d$. Here, natural density means the density calculated with respect to the sequence $A_n=[-n,n]^d$, note that this set can have a different density with respect to other sequences.

In particular, the visible sets of $\mathbb Z^2$, given by $$V=\{ (n,m) \in \mathbb Z^2 : \mbox{gcd}(m,n) =1 \}$$ have natural density $\frac{1}{\zeta(2)}$.

The so called "cut and project" formalism establises a connection between the above example and some sets in compact groups, which appeared in my research area recently.

Consider the group $$\mathbb K:= \prod_{p \in P} \left( \mathbb Z^2 / p \mathbb Z^2 \right)$$ where $p$ denotes the set of all primes. $\mathbb K$ is a compact Abelian group, and hence has a probability Haar measure $\theta_{\mathbb K}$.

Now, $\phi(m,n) := \left( (m,n)+p \mathbb Z^2 \right)_{p \in P}$ defines an embedding of $\mathbb Z^2$ into $\mathbb K$.

Define the set $$W:= \prod_{p \in P} \left( \bigl(\mathbb Z^2 / p \mathbb Z^2\bigr) \backslash \{ (0,0) + p\mathbb Z^2 \} \right)$$

Then, the visible points of $\mathbb Z^2$ are exactly $$V= \phi^{-1}(W)$$

The set $W$, which is used in the study of diffraction of $V$, has the following properties:

$W$ is closed and hence compact.

is closed and hence compact. $W$ has empty interior (hence is fractal shape).

has empty interior (hence is fractal shape). $\theta_{\mathbb K}(W) = \frac{1}{\zeta(2)}$

The last property is where I was going to, and it is intuitively not that hard to see once you identify $\theta_{\mathbb K}(W)$ as the product of the counting measures on $\mathbb Z^2 / p \mathbb Z^2$: this immediatelly gives $$\theta_{\mathbb K}(W) = \prod_{p \in P}\frac{p^2-1}{p^2}$$

P.S. There are similar appearences of $\zeta(n)$ in the study of $k$th power free integers, that is all the integers $n \in \mathbb Z$ which are not divisible by the $k$th power of any prime, for a fixed $k$.