Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$.
Say that a point $(x,y)$ of $Q$ is *visible from the origin* if the
segment from $(0,0)$ to $(x,y) \in Q$ passes through no other point of $Q$.
So points block visibility, and the only points visible from the origin are those with
a clear line of sight:

Let $\nu(n)$ be the ratio of the number of points visible within the
square with corner $(n,n)$ to $n^2$. For example, for $n=8$, $43$ of the $64$ points
are visible, so $\nu(n) = 43/64 \approx 0.67$.

Q: What is $\lim_{n \to \infty} \nu(n)$?

It appears to be approaching $\approx 0.614$:

The question bears some similarity to Polya's Orchard Problem
(T.T. Allen, "Polya's orchard problem,"
*The American Mathematical Monthly*
93(2): 98-104 (1986). Jstor link), but I cannot see my question
is answered in that literature.

One could ask the same question for $\mathbb{Z}^d$.

**Answered**by Pete Clark: the limit is $\frac{1}{\zeta(d)}$. Thus in higher dimensions, almost all the lattice points are visible (because $\zeta(d)$ approaches $1$).

lattice points from originis also an easy way to find the answer. It links to: shreevatsa.wordpress.com/2008/11/07/… $\endgroup$