Problem 522

Erdős Problem #522: declared status 'open'. Formalized: yes. Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$. Is it true that, if $R_n$ is the number of roots of $f(z)$ in $\{ z\in \mathbb{C} : \lvert z\rvert \leq 1\}$, then$$\frac{R_n}{n/2}\to 1$$almost surely? Prize: no. Tags: analysis, polynomials, probability.