Zeros distribution of gaussian entire functions

In this paper we consider a random entire function of the form $f(z,\omega )=\sum\nolimits_{n=0}^{+\infty}\xi_n(\omega )a_nz^n,$where$\xi_n(\omega )$areindependentstandard\text{break}complexgaussianrandomvariablesand$a_n\in$\mathbb{C}$$satisfytherelations\text{break}$\varlimsup\limits_{n\to+\infty}\sqrt[n]{|a_n|}=0$and$ #{n\colon a_n\neq0}=+\infty.$Weinvestigateasymptoticpropertiesoftheprobability$P_0®=P{\omega\colon f(z,\omega )$hasnozerosinside$r$\mathbb{D}$}.$Denote$ p_0®=\ln^-P_0®,\ N®=#{n\colon \ln (|a_n|r^n)>0},$ $s(r)={\sum }_{n=0}^{+\mathrm{\infty }}{\ln }^{+}(\mathrm{\mid }{a}_n\mathrm{\mid }{r}^n)\mathrm{.}$ Assuming that $a_0\mathrm{\ne }0$ we prove that $ 0\leq\varliminf_{r\to+\infty,\ r\notin E}\frac{\ln(p_0®- s®)}{\ln s®},\ \varlimsup_{r\to+\infty,\ r\notin E}\frac{\ln(p_0®- s®)}{\ln s®}\leq\frac12, $ $\underset{r\to +\mathrm{\infty },r\mathrm{\notin }E}{\lim }\frac{\ln (p_0(r)-s(r))}{\ln N(r)}=1.$ where $E$ is a set of finite logarithmic measure. Remark that the previous inequalities are sharp. Also we give an answer to open question from \cite[p. 119]{nishry 5}.