Geodesic
On the expected number of real zeros of random polynomials I.~Coefficients with zero means
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 229-248

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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 $Q_n$. 
@article{TVP_1971_16_2_a2,
     author = {I. A. Ibragimov},
     title = {On the expected number of real zeros of random polynomials {I.~Coefficients} with zero means},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {229--248},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a2/}
}
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I. A. Ibragimov. On the expected number of real zeros of random polynomials I.~Coefficients with zero means. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 229-248. http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a2/