The angular distribution of zeros of random analytic functions

Abstract

It is proved, that for the majority (in the sense of probability measure) of functions $f$, analytic in the unit disk with unbounded Nevanlinna characteristic $T_f(r)$, and for all $\alpha<\beta\le\alpha+2\pi$ the relation $$ N_f(r,\alpha,\beta,0)\sim\frac{\beta-\alpha}{2\pi}T_f(r),\quad r\to1, $$ holds, where $N_f(r,\alpha,\beta,0)$ is the integrated counting functions of zeros of $f$ in the sector $\{z\in\mathbb C\colon\ 0<|z|\le r,\ \alpha\le\arg_\alpha z<\beta\}$. The analogous proposition is obtained for entire functions under some conditions on their growth.