On distribution of zeros of random polynomials in complex plane

Abstract

Let $G_n(z)=\xi_0+\xi_1z+...+\xi_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2\pi]$ asymptotically as $n\to\infty$. We also prove that the condition $\E\ln(1+|\xi_0|)<\infty$ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.