Paper detail

On distribution of zeros of random polynomials in complex plane

Let $G_n(z)=ξ_0+ξ_1z+...+ξ_n z^n$ 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 $[0,2π]$ 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.