  Previously Heyman and Shparlinski gave an asymptotic formula with error term
for the number of Eisenstein polynomials of fixed degree and bounded height.
Let $\psi(f)$ denote the number of primes for which a polynomial $f$ is
Eisenstein. We give expressions for the mean and variance of the function
$\psi$ for each fixed degree, where the polynomials are ordered according to
their height.

Number Theory