Let $p_n$ be the n-th prime number and $c_n$ be the n-th composite number. Prove that

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

The beauty of the above result is that the first limit is a series over prime and the other is a series over composites which are exactly what primes are not, yet they converge to the same limit. The same result hold if the sequence of primes (or composites) are replaced by the sequence of natural number. Yet the two limits are equal. This is a specific example of a general family of results of this kind.

The question is to not merely to prove the above result but 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.

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.