Squarefree values of random polynomials are Gaussian mixtures

Abstract

Fix $d\ge1$ 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 $n\leq x$. In the range $x=H^{o\left(1\right)}$, we prove that $E_{P}\left(x\right)/x^{\frac{1}{4}}$ converges to a Gaussian mixture that is not itself Gaussian, $\sqrt{C\left(G\right)}Z$, where $G$ is the weak limit, as $H\rightarrow\infty$, of the prime-indexed vector $\left(P\bmod p^{2}\right)_{p}$, $C$ is a nonnegative function and $Z$ is an independent standard Gaussian. This answers a question of Sofos. Finally, in the appendix, we establish that $\log \mathbb{E}[C(G)^m]\asymp_d m^2/(\log m)^2$ as $m\to\infty$.