Let $F(x,y)$ be an irreducible binary form with integer coefficients and degree $d \geq 3$. Denote by $A_F$ the area of the region
$$\displaystyle \{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq 1\}.$$
It is known that $A_F < \infty$ (in fact, uniformly bounded) for all binary forms $F$, even though the region is not in general compact. Put
$$\displaystyle N_F(Z) = \# \{(x,y) \in \mathbb{Z}^2 : |F(x,y)| \leq Z\}.$$
Mahler proved in 1933 that
$$\displaystyle N_F(Z) = A_F Z^{\frac{2}{d}} + O_F\left(Z^{\frac{1}{d-1}} \right).$$
My question is as follows. Define
$$\mathcal{N}_F(Z) = \# \{(x,y) \in \mathbb{P}^2 : |F(x,y)| \leq Z \}$$
where $\mathbb{P}$ denotes the set of prime numbers (with negative signs allowed). Then is it true that Mahler's theorem holds for $\mathcal{N}_F(Z)$, adjusted appropriately for primes? Namely, do there exist local factors $\mu_p$ such that
$$\displaystyle \mathcal{N}_F(Z) \sim A_F \prod_p \mu_p \cdot d^2 Z^{\frac{2}{d}} (\log Z)^{-2}?$$
The term $d^2 Z^{2/d} (\log Z)^{-2}$ comes from the fact that there are roughly $\displaystyle 4\left(\frac{Z^{1/d}}{\log Z^{1/d}}\right)^2$ many pairs of primes in the box $[-Z^{1/d}, Z^{1/d}]$.
Any help would be appreciated.
