Squarefree values of random polynomials are Gaussian mixtures

Fix $d\ge 1$ and choose a polynomial $P$ uniformly at random from the set of integer polynomials of degree $d$ whose coefficients have absolute value at most $H$. We study $E_P$, the error in the count of squarefree values $P(n)$ for $n\le x$. In the range $x=H^{o\left(1\right)}$, we prove that $E_P\left(x\right)\mathrm{/}{x}^{\frac{1}{4}}$ converges to a Gaussian mixture that is not itself Gaussian, $\sqrt{C\left(G\right)}Z$, where $G$ is the weak limit, as $H\to \mathrm{\infty }$, of the prime-indexed vector ${(P{p}^2)}_p$, $C$ is a nonnegative function and $Z$ is an independent standard Gaussian. This answers a question of Sofos. Finally, in the appendix, we establish that $\log \mathbb{E}[C(G{)}^m]{\asymp }_d{m}^2\mathrm{/}(\log m{)}^2$ as $m\to \mathrm{\infty }$.