Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are finitely many number fields $K$ below a given discriminant bound $|D_{K/\mathbb{Q}}| \leq X$. Let $\mathcal{S}(X)$ be that finite set.

Question. What is a good guess for the order of growth of $\# \mathcal{S}(X)$ as a function of $X$? Any reasons to expect (or to doubt) that this count should be asymptotic to a regular (say, smooth) function of $X$?

If so, what about finer statistics of number fields at large (unconstrained degrees with $|D_{K/\mathbb{Q}}|$ as the sole parameter), such as the function of typical degree $d(X) := \frac{1}{\#\mathcal{S}(X)} \sum_{K \in \mathcal{S}(X)} [K:\mathbb{Q}]$, the mean Euler-Kronecker invariant $\gamma_K$, or the mean minimum height of a generator of a random number field, $$ h := \lim_{X \to \infty} \frac{1}{\#\mathcal{S}(X)}\sum_{K \in \mathcal{S}(X)} \min_{\alpha: \, K = \mathbb{Q}(\alpha)} h(\alpha) \in [0,\infty]. $$ Assuming this extended number $h$ exists, should it be $0$, $\infty$, or a positive finite number? The former alternative would mean that when number fields are ordered by discriminant, a random number field can be generated by a root of an integer polynomial whose coefficients are asymptotically sub-exponential in the degree. The second alternative would mean that a random field requires more than an exponentially large coefficients for all of its generating elements. While the third alternative says that a random number field has the Bogomolov height gap property.

Since the discriminant of a number field $K \neq \mathbb{Q}$ is bounded from below by an exponential of the degree $[K:\mathbb{Q}]$, for instance by Minkowski's Geometry of Numbers bound, there are finitely many number fields $K$ below a given discriminant bound $|D_{K/\mathbb{Q}}| \leq X$. Let $\mathcal{S}(X)$ be that finite set.

Question. What is a good guess for the order of growth of $\# \mathcal{S}(X)$ as a function of $X$? Any reasons to expect (or to doubt) that this count should be asymptotic to a regular (say, smooth) function of $X$?

If so, what about finer statistics of number fields at large (unconstrained degrees with $|D_{K/\mathbb{Q}}|$ as the sole parameter), such as the function of typical degree $d(X) := \frac{1}{\#\mathcal{S}(X)} \sum_{K \in \mathcal{S}(X)} [K:\mathbb{Q}]$, the mean Euler-Kronecker invariant $\gamma_K$, or the mean minimum height of a generator of a random number field, $$ h := \lim_{X \to \infty} \frac{1}{\#\mathcal{S}(X)}\sum_{K \in \mathcal{S}(X)} \min_{\alpha: \, K = \mathbb{Q}(\alpha)} h(\alpha) \in [0,\infty]. $$ Assuming this extended number $h$ exists, should it be $0$, $\infty$, or a positive finite number? The former alternative would mean that when number fields are ordered by discriminant, a random one can be generated by a root of an integer polynomial whose coefficients are asymptotically sub-exponential in the degree. The second alternative would mean that a typical number field is complicated to describe, in the sense of requiring super-exponentially large coefficients to write down a generating element. The third alternative says simply that typical number fields have Bogomolov's height gap property.