Let $D_4$ denote the (isomorphism class of) dihedral group of 8 elements. For a given polynomial $f$ with integer coefficients, define $\text{Gal}(f)$ to be the Galois group of the Galois closure of $f$ and $\Delta(f)$ to be the discriminant of $f$. Consider the counting function

$$\displaystyle N(f; D_4, Z) = \#\{f(x) \in \mathbb{Z}[x] : \deg f = 4, |\Delta(f)| < Z, \text{Gal}(f) \cong D_4 \}.$$

Is there a known asymptotic formula for $N(f; D_4, Z)$, or at least an asymptotic lower bound expected to be of the correct order of magnitude? 

There are several related results. For example, if one is to count the number of *quartic fields* with Galois group $D_4$ instead, then it was shown by Cohen, Diaz y Diaz, and Olivier that there are roughly $cZ$ many such fields with absolute discriminant up to $Z$, where $c \approx 0.0523\cdots$. One should note that the density of quartic fields with Galois group $D_4$ among all quartic fields is positive, as was shown by Bhargava. This is in contrast to the case with polynomials, where $100\%$ of all quartic polynomials have Galois group $S_4$. It was further shown by Bhargava and Shankar that there are at most $O_{\varepsilon}\left(Z^{\frac{3}{4} + \epsilon}\right)$ number of quartic polynomials of "naive height" up to $Z$, where as the number of quartic polynomials of naive height up to $Z$ is $Z^{\frac{5}{6}}$. The naive height is defined using the $I,J$ invariants of quartic polynomials and is designed to mimic the discriminant but is easier to deal with. The order of magnitude should be the same for both counting methods.