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 $H \to \infty$?