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The limiting ratio is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}= 0.6079271018540266286632767792\ldots$

The question is easily seen to be equivalent to "What is the probability that two integers are relatively prime?" This old chestnut of elementary number theory has been addressed before on this site: see here. The idea is incredibly simple and appealing: we consider the events "$a$ and $b$ are both divisible by $p$" -- as $p$ runs over all primes -- as independent and use the Euler product for $\zeta(2)$. It is not completely obvious how to make this reasoning rigorous, but it can be and has been done in any number of ways. The linked to answer contains one of them.

Since the OP is highly involved in discrete geometry I wanted to mention a more general result: if $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ is a bounded, Jordan measurable region, then the number of primitive (= visible from the origin) lattice points in the dilate $r\Omega$ of $\Omega$ is asymptotic to $\left(\frac{\operatorname{Vol} \Omega}{\zeta(n)} \right) r^n$. For $n =2$ this is proved for instance in $\S$ 24.10 (of the sixth edition, but I don't think that matters) of Hardy and Wright's An Introduction to the Theory of Numbers. It is stated in $\S$ 7.3 of these notes and placed there in a more general context: namely it is related to the celebrated Minkowski-Hlawka Theorem. (The proof was left to two students in the graduate seminar I was then running. They presented a proof, but I didn't get around to incorporating their arguments into the writeup. Anyway the transition from the two-dimensional case is straightforward.)

The limiting ratio is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}= 0.6079271018540266286632767792\ldots$

The question is easily seen to be equivalent to "What is the probability that two integers are relatively prime?" This old chestnut of elementary number theory has been addressed before on this site: see here. The idea is incredibly simple and appealing: we consider the events "$a$ and $b$ are both divisible by $p$" -- as $p$ runs over all primes -- as independent and use the Euler product for $\zeta(2)$. It is not completely obvious how to make this reasoning rigorous, but it can be and has been done in any number of ways. The linked to answer contains one of them.

Since the OP is highly involved in discrete geometry I wanted to mention a more general result: if $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ is a bounded, Jordan measurable region, then the number of primitive (= visible from the origin) lattice points in the dilate $r\Omega$ of $\Omega$ is asymptotic to $\left(\frac{\operatorname{Vol} \Omega}{\zeta(n)} \right) r^n$. For $n =2$ this is proved for instance in $\S$ 24.10 (of the sixth edition, but I don't think that matters) of Hardy and Wright's An Introduction to the Theory of Numbers. It is stated in $\S$ 7.3 of these notes and placed there in a more general context: namely it is related to the celebrated Minkowski-Hlawka Theorem. (The proof was left to two students in the graduate seminar I was then running. They presented a proof, but I didn't get around to incorporating their arguments into the writeup. Anyway the transition from the two-dimensional case is straightforward.)

The limiting ratio is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}= 0.6079271018540266286632767792\ldots$

The question is easily seen to be equivalent to "What is the probability that two integers are relatively prime?" This old chestnut of elementary number theory has been addressed before on this site: see here. The idea is incredibly simple and appealing: we consider the events "$a$ and $b$ are both divisible by $p$" -- as $p$ runs over all primes -- as independent and use the Euler product for $\zeta(2)$. It is not completely obvious how to make this reasoning rigorous, but it can be and has been done in any number of ways. The linked to answer contains one of them.

Since the OP is highly involved in discrete geometry I wanted to mention a more general result: if $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ is a bounded, Jordan measurable region, then the number of primitive (= visible from the origin) lattice points in the dilate $r\Omega$ of $\Omega$ is asymptotic to $\left(\frac{\operatorname{Vol} \Omega}{\zeta(n)} \right) r^n$. For $n =2$ this is proved for instance in $\S$ 24.10 (of the sixth edition, but I don't think that matters) of Hardy and Wright's An Introduction to the Theory of Numbers. It is stated in $\S$ 7.3 of these notes and placed there in a more general context: namely it is related to the celebrated Minkowski-Hlawka Theorem. (The proof was left to two students in the graduate seminar I was then running. They presented a proof, but I didn't get around to incorporating their arguments into the writeup. Anyway the transition from the two-dimensional case is straightforward.)

The limiting ratio is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}= 0.6079271018540266286632767792\ldots$

The question is easily seen to be equivalent to "What is the probability that two integers are relatively prime?" This old chestnut of elementary number theory has been addressed before on this site: see here. The idea is incredibly simple and appealing: we consider the events "$a$ and $b$ are both divisible by $p$" -- as $p$ runs over all primes -- as independent and use the Euler product for $\zeta(2)$. It is not completely obvious how to make this reasoning rigorous, but it can be and has been done in any number of ways. The linked to answer contains one of them.

Since the OP is highly involved in discrete geometry I wanted to mention a more general result: if $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ is a bounded, Jordan measurable region, then the number of primitive (= visible from the origin) lattice points in the dilate $r\Omega$ of $\Omega$ is asymptotic to $\left(\frac{\operatorname{Vol} \Omega}{\zeta(n)} \right) r^n$. For $n =2$ this is proved for instance in $\S$ 24.10 (of the sixth edition, but I don't think that matters) of Hardy and Wright's An Introduction to the Theory of Numbers. It is stated in $\S$ 7.3 of these notes and placed there in a more general context: namely it is related to the celebrated Minkowski-Hlawka Theorem. (The proof was left to two students in the graduate seminar I was then running. They presented a proof, but I didn't get around to incorporating their arguments into the writeup. Anyway the transition from the two-dimensional case is straightforward.)

The limiting ratio is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}$. A proof can be found for instance in $\S$ 24.10 (of the sixth edition, but I don't think that matters) of Hardy and Wright's An Introduction to the Theory of Numbers. See $\S$ 7.3 of these notes for the statement of a generalization to the primitive lattice point enumerator for regions in $n$-dimensional Euclidean space.  (The proof was left to two students in the graduate seminar I was then running. They presented a proof, but I didn't get around to incorporating their arguments into the writeup. Anyway the transition from the two-dimensional case is straightforward.)

This is a standard result in the elementary Geometry of Numbers, equivalent to the problem in elementary number theory asking for the probability that two positive integers are relatively prime. You can see intuitively that the above answer is correct by considering the events "$a$ and $b$ are both divisible by $p$" -- as $p$ runs over all primes -- as independent and using the Euler product for $\zeta(2)$.

Added: In $n$-dimensional space the same reasoning leads to the answer is $\frac{1}{\zeta(n)}$. (Which of course means that it is in some sense unknown exactly what the answer is whenever $n$ is odd...)

The limiting ratio is $\frac{1}{\zeta(2)} = \frac{6}{\pi^2}= 0.6079271018540266286632767792\ldots$

The question is easily seen to be equivalent to "What is the probability that two integers are relatively prime?" This old chestnut of elementary number theory has been addressed before on this site: see here. The idea is incredibly simple and appealing: we consider the events "$a$ and $b$ are both divisible by $p$" -- as $p$ runs over all primes -- as independent and use the Euler product for $\zeta(2)$. It is not completely obvious how to make this reasoning rigorous, but it can be and has been done in any number of ways. The linked to answer contains one of them.

Since the OP is highly involved in discrete geometry I wanted to mention a more general result: if $n \geq 2$ and $\Omega \subset \mathbb{R}^n$ is a bounded, Jordan measurable region, then the number of primitive (= visible from the origin) lattice points in the dilate $r\Omega$ of $\Omega$ is asymptotic to $\left(\frac{\operatorname{Vol} \Omega}{\zeta(n)} \right) r^n$. For $n =2$ this is proved for instance in $\S$ 24.10 (of the sixth edition, but I don't think that matters) of Hardy and Wright's An Introduction to the Theory of Numbers. It is stated in $\S$ 7.3 of these notes and placed there in a more general context: namely it is related to the celebrated Minkowski-Hlawka Theorem. (The proof was left to two students in the graduate seminar I was then running. They presented a proof, but I didn't get around to incorporating their arguments into the writeup. Anyway the transition from the two-dimensional case is straightforward.)