2014/01/13 by A. O. Kuryliak, Kuryliak, A. O., О. Б. Скасків +2
Mathematics · #30B20 #30D35 #30E15 #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #math.CV #msc:30B20 #msc:30D35 #msc:30E15
paper · pdf · doi:10.48550/arxiv.1401.2776
arxiv created 2014/01/13 · openalex publication_date 2014/01/13 · arxiv updated 2014/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we consider a random entire function of the form f(z,ω)=∑\nolimitsn=0+∞ξn(ω)anzn, where ξn(ω) are independent standard\break complex gaussian random variables and an∈ℂ satisfy the relations\break \varlimsupn→+∞√[n]|an|=0 and #\n\colon an≠0\=+∞. We investigate asymptotic properties of the probability P0(r)=P\ω\colon f(z,ω) has no zeros inside r\mathbbD\. Denote p0(r)=ln-P0(r), N(r)=#\n\colon ln (|an|rn)>0\, s(r)=∑n=0+∞ln+(|an|rn). Assuming that a0≠0 we prove that 0≤\varliminfr→+∞, r∉ E(ln(p0(r)- s(r)))/(ln s(r)), \varlimsupr→+∞, r∉ E(ln(p0(r)- s(r)))/(ln s(r))≤\frac12, limr→+∞, r∉ E(ln(p0(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.