Let $p_n$ be the n-th prime number and $c_n$ be the n-th composite number. We have

$$ \lim_{n \to \infty}\frac{1}{n} \sum_{r=1}^{n}\frac{p_n^2}{p_n^2 + p_r^2} = \lim_{n \to \infty}\frac{1}{n} \sum_{r=1}^{n}\frac{c_n^2}{c_n^2 + c_r^2} = \frac{\pi}{4}. $$

The beauty of the above result is that the first limit is a series over prime and the other is a series over composites. Similar results hold if the sequence of primes (or composites) are replaced by the sequence of natural number. This is a specific example of a general family of results of this kind.

The question is understand why such a relationships hold. I have been working on this and wanted to share with and learn from other number theorists. Has anyone seen similar results in mathematics literature? Any reference would be helpful.