Consider the function $Pt(n)$. It tells us how many primitive Pythagorean Triples there are (below $n$) when any argument $n \in \mathbb{N}$ is plugged in. Is there an 'exact formula'; i.e. an elementary function of even a combination of known special functions like the Gamma and Error Function, that describes $Pt(n)$ ?
Max
Edit: I'm also interested in the exact value of the limit of $Pt(n)/n$ when $n$ tends to infinity.
To answer the sharpened version of the question I suggested (the number of primitive Pythagorean triples with largest element ${\lt}n$ ): by the parametrization of pythagorean triples as sums of two squares, this is (essentially) equal to the question of how many ways there are of expressing all the odd numbers ${\lt}n$ as a sum of two squares. Mathworld's page on the sum-of-two-squares function at http://mathworld.wolfram.com/SumofSquaresFunction.html indicates that this is proportional to $n$ (though it might take some work to explicitly work out the constant of proportionality for the odd case), and so in fact your intuition is wrong; the limit you suggest tends to a finite positive value.