I have been working on a problem concerning the "line of sight" from a fixed integer co-ordinate — let's say $(0,0)$ — to a variable co-ordinate $(a,b)$. Having a line of sight means that there are no other integer co-ordinates on a straight line connecting the two points.


This is a a well documented problem and there isn't very much that's worth exploring on it. However, when investigating the long run behaviour, I noticed that if were look at the probability that a uniformly chosen point has a line of sight as the co-ordinate approaches infinity (I define this more formally below), then the proof that this probability approaches $0$ relies on the fact that the probability that these the $x$ and $y$ co-ordinates are co-prime also approaches $0$. This is a fact that is clearly true (at least intuitively), however, I have struggled to construct (or find) a proof of this fact.


More formally, my question asks the following:



> 
> Prove that for $a,b \sim \operatorname{Uniform}\{n, \space n+1, \dotsc \space 2n\}$, the following is true:
> 
> 
> $$ \mathbb{P}(\{ \text{$a$ and $b$ are co-prime} \}) \rightarrow0$$
> as $n \rightarrow \infty$.
> 
> 
> 


As stated above, this is clearly true. What I am after is either a proof or a reference to a proof of the above claim so that I can justify this result rigorously.


I would be grateful for any contributions here.




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Note: as discussed in the comments, I am aware of the fact that for any two randomly chosen positive integers, the probability that they are coprime will be $6 / \pi^2$ [(see proof here)](https://math.stackexchange.com/questions/64498/probability-that-two-random-numbers-are-coprime-is-frac6-pi2), however, the application of this to my question is not immediately clear me to beyond providing me with the intuition for my conjecture.



Proving that $P($$\{\text{$a$ and $b$ are co-prime}$ }$)=0$ for $a,b$ following the Uniform distribution over $[n, 2n]$ as $n \rightarrow \infty$
I seem to get the probability is still the usual $1/\zeta(2)$ by the usual inclusion/exclusion argument, but possibly there's a mistake in the following calculation:
$$
\begin{aligned}
\frac{1}{n^2}\sum\_{a,b\in[n,2n]} \;\;\sum\_{d\mid(a,b)} \mu(d)
&= \frac{1}{n^2}
\sum\_{1\le d\le 2n} \mu(d)\sum\_{\substack{A,B\\ A\in[n/d,2n/d]
\\ B\in[n/d,2n/d] \\}} 1 \quad\text{setting $a=Ad$ and $b=Bd$,}\\
&= \frac{1}{n^2}\mu(d) \sum\_{1\le d\le 2n} \left(\frac{n}{d} + O(1)\right)^2 \\
&= \frac{1}{n^2} \sum\_{1\le d\le 2n} \mu(d)\frac{n^2}{d^2} + O\left(\frac{n}{d}\right) \\
&= \sum\_{1\le d\le 2n} \mu(d)\frac{1}{d^2}
+ O\left( \frac{1}{n} \sum\_{1\le d\le 2n}\frac{1}{d}\right) \\
&= \frac{1}{\zeta(2)} + O\left(\frac{\log n}{n}\right).
\end{aligned}
$$


