A primitive Pythagorean triplet is a triplet $a^2 + b^2 = c^2$ be where $a,b,c$ have no common factors and is generated by $a = x^2 - y^2, b = 2xy, c = x^2 + y^2$ where $x > y, \gcd(x,y) = 1$. My experimental data suggests that
$$ \frac{1}{5} + \frac{1}{13} + \frac{1}{17} + \frac{1}{25} + \frac{1}{29} + \frac{1}{37} + \cdots + \frac{1}{x^2 + y^2} \sim \frac{\pi}{6}\log x$$
where the summation is taken over the hypotenuse of all primitive right triangles such that $y < x$. Can this be proved or disproved?
Related question: What is the sum of the reciprocal of the square of hypotenuse of Pythagorean triangles?