Erdős Problem #990

declared status 'disproved'. Formalized: no. Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial. Is it true that, if $f$ has roots $z_1,\ldots,z_d$ with corresponding arguments $\theta_1,\ldots,\theta_d\in [0,2\pi]$, then for all intervals $I\subseteq [0,2\pi]$$$\left\lvert (\# \theta_i \in I) - \frac{\lvert I\rvert}{2\pi}d\right\rvert \ll \left(n\log M\right)^{1/2},$$where $n$ is the number of non-zero coefficients of $f$ and$$M=\frac{\lvert a_0\rvert+\cdots +\lvert a_d\rvert}{(\lvert a_0\rvert\lvert a_d\rvert)^{1/2}}.$$ Current best: Erd\H{o}s and Tur\'{a}n [ErTu50] proved such an upper bound with $n$ replaced by $d$. Prize: no. Tags: analysis.

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