On distribution of zeros of random polynomials in complex plane

Abstract

Let $G_n(z)={\xi }_0+{\xi }_{1}z+...+{\xi }_n{z}^n$ 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 $\E\ln(1+|ξ_0|)<\infty$ is necessary and sufficient for the roots to asymptotically concentrate near the unit circumference.