Entire Gaussian Functions: Probability of Zeros Absence

Abstract

In this paper, we consider a random entire function of the form f(z,ω)=∑n=0+∞εn(ω1)×ξn(ω2)fnzn, where (εn) is a sequence of independent Steinhaus random variables, (ξn) is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers fn∈C is such that lim¯n→+∞|fn|n=0 and #{n:fn≠0}=+∞. We investigate asymptotic estimates of the probability P0(r)=P{ω:f(z,ω) has no zeros inside rD} as r→+∞ outside of some set E of finite logarithmic measure, i.e., ∫E∩[1,+∞)dlnr<+∞. The obtained asymptotic estimates for the probability of the absence of zeros for entire Gaussian functions are in a certain sense the best possible result. Furthermore, we give an answer to an open question of A. Nishry for such random functions.