Affiliation:
1. Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine
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.
Subject
Geometry and Topology,Logic,Mathematical Physics,Algebra and Number Theory,Analysis
Reference29 articles.
1. On the distribution of zeros and a-values of a random integral function;Littlewood;Ann. Math. (II),1948
2. Littlewood, J.E. (1982). Collected Papers, Oxford University Press. 1948, 2 Vols.
3. The distribution of the values of an entire function whose coefficients are independent random variables;Offord;Proc. Lond. Math. Soc.,1965
4. The distribution of zeros of series whose coefficients are independent random variables;Offord;Indian J. Math.,1967
5. The distribution of the values of a random functions in the unit disk;Offord;Stud. Math.,1972
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献