Entire Gaussian Functions: Probability of Zeros Absence.

In this paper, we consider a random entire function of the form f (z , ω) = ∑ n = 0 + ∞ ε n (ω 1) × ξ n (ω 2) f n z n , where (ε n) is a sequence of independent Steinhaus random variables, (ξ n) is the a sequence of independent standard complex Gaussian random variables, and a sequence of numbers f n ∈ C is such that lim ¯ n → + ∞ | f n | n = 0 and # { n : f n ≠ 0 } = + ∞. We investigate asymptotic estimates of the probability P 0 (r) = P { ω : f (z , ω) has no zeros inside r D } as r → + ∞ outside of some set E of finite logarithmic measure, i.e., ∫ E ∩ [ 1 , + ∞) d ln r < + ∞ . 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. [ABSTRACT FROM AUTHOR]