  We establish the central limit theorem for the number of real roots of the
Weyl polynomial $P_n(x)=xi_0 + xi_1 x+ ... + xi_n (n!)^{(-1/2)} x^n$, where
$xi_i$ are iid Gaussian random variables. The main ingredients in the proof are
new estimates for the correlation functions of the real roots of $P_n$ and a
comparison argument exploiting local laws and repulsion properties of these
real roots.
