Problem 521

Erdős Problem #521: declared status 'open'. Formalized: no. Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k$ then is it true that, almost surely,$$\lim_{n\to \infty}\frac{R_n}{\log n}=\frac{2}{\pi}?$$ Prize: no. Tags: analysis, polynomials, probability.