  Roots of random polynomials have been studied exclusively in both analysis
and probability for a long time. A famous result by Ibragimov and Maslova,
generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the
expectation of the number of real roots is $\frac{2}{\pi} \log n + o(\log n)$.
In this paper, we determine the true nature of the error term by showing that
the expectation equals $\frac{2}{\pi}\log n + O(1)$. Prior to this paper, such
estimate has been known only in the gaussian case, thanks to works of Edelman
and Kostlan.
