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Growth Rate of Zeros of Almost all Random Entire Functions
时间  Datetime
2018-12-04 10:00 — 11:00 
地点  Venue
Middle Lecture Room
报告人  Speaker
Zhuan Ye
单位  Affiliation
Department of Mathematics and Statistics University of North Carolina University Wilmington
邀请人  Host
姚卫红
报告摘要  Abstract

Let f(z, ω) = P∞ j=1 Aj (ω)z j be a random entire function, where Aj (ω) are independent and identically distributed random variables defined on a probability space (?, F, µ). If Aj ’s are either equal to e 2πiθj (ω) with θj (ω) being of standard uniform distribution or Gaussian random variables with standard Gaussian distribution, then we prove that for almost all f(z, ω) and any a ∈ C with f(0, ω) − a 6= 0, there is r0 such that when r > r0, 1 2π Z 2π 0 log+ |f(reit, ω)| dt = Z r 0 n(t, ω, a, f) t dt + log |f(0, ω) − a| + O(1), where n(t, ω, a, f) is the number of zeros of f(z, ω)−a in |z| < t and O(1) is independent of ω. The identity can be regarded as the Nevanlinna’s second main theorem and has improved some previous theorems on the growth rate of zeros of random entire functions.