On the expected number of real zeros of random polynomials I. Coefficients with zero means

Let $\xi_j$, $j=0,1,\dots$, be independent identically distributed random variables with $\mathbf E\xi_j=0$ and belong to the domain of attraction of the normal law.
The main result is:
$$ \mathbf E\{N_n\mid Q_n(x)\not\equiv0\}\underset{n\to\infty}\sim\frac2\pi\ln n\quad\text{if }\mathbf P\{\xi_j\ne0\}>0 $$
where $Q_n(x)=\sum_{j=0}^n\xi_jx^j$, $N_n$ is the number of real roots of $Q_n$.