Let $R$ be a semi-algebraic, compact region in $\mathbb{R}^n$ with positive Lebesgue measure. Let $N(R) = \# (R \cap \mathbb{Z}^n)$. Davenport's lemma asserts that we have

$$\displaystyle N(R) = \operatorname{Vol}(R) + O(\max\{\operatorname{Vol} \overline{R}\}),$$

where the maximum is taken over all projections $\overline{R}$ of $R$ onto coordinate subspaces obtained by setting some coordinates to zero. Is there an analogous statement for

$$\displaystyle N'(R) = \# (R \cap \mathfrak{P}^n),$$

where $\mathfrak{P}$ denotes the set of primes and their negatives (so that $N'(R)$ counts the number of lattice points in $R$ whose coordinates are all primes or the negative of a prime)?