Let me call a pair of integers $a, b$ *acceptable* if the equation $ax^2 + by^2 = z^2$ has a non-trivial rational solution. Theorem 4.5.4 of Cojocaru-Murty's book on Sieves says that the number of acceptable pairs of integers $a, b$ with $1 \leq a, b \leq H$ is $\ll H^2 / \log \log H$. This is proved using the Turan sieve. Serre had previously proved in a short paper in 1990 that the estimate can be improved to $H/(\log H)^\delta$ for some $\delta >0$. This proof of the latter result uses the large sieve. 


Question. Is it possible to give an asymptotic formula for the number of acceptable pairs of integers $a, b$ with $1 \leq a, b \leq H$ as $H \to \infty$? 



Hilbert symbol averages
This problem has been considered by a few authors. Versions for $a$ and $b$ *rational* were considered by Hooley and Guo (independently). See 


Hooley - On ternary quadratic forms that represent zero.


Guo - On solvability of ternary quadratic forms.


Hooley obtained the (sharp) lower bound of the form $H^3/(\log H)^{3/2}$ for the corresponding counting problem. Guo obtained an asymptotic formula for the slightly different problem where $a$ and $b$ are assumed to be square-free (i.e. the numerators and denominators are square-free).


An asymptotic formula without the square-free assumption does not seem to be known.


More recent work:


Friedlander, Iwaniec - Ternary quadratic forms with rational zeros


considers the case where $a$ and $b$ are integers, and obtained an asymptotic formula of the order $H^2/(\log H)$ under the additional assumption that $a$ and $b$ are odd, coprime and square-free. An asymptotic formula for general $a$ and $b$ might be possible to obtain from their methods, but probably quite messy.


