Problem
Define a sequence $a$ with $a_i=\sum_{j=1}^{n}\sum_{k=1}^{n}[gcd (j,k)=i]$. We define another sequence $b$ about $a$, $b_i=\frac{a_i}{\sum_{j=1}^{n}a_i}$. In $n \to \infty$, what characteristics does the sequence $b$ exhibit as a whole?
My attempt
The following three sets of images show the first $10$ numbers of $b$ in $n=10, n=100, n=10000$ respectively.
This looks like the $b$ series tends to some fixed number when $n \to \infty$. But I don't know why.
As $n\to \infty$ you have $b_i \to \frac{6}{\pi^2 i^2}$, from the density of coprime pairs and the fact that a pair $a, b\leq n$ with gcd $i$ is equivalent to a coprime pair $a,b\leq \frac ni$.