Growth Rate of Zeros of Almost all Random Entire Functions


Zhuan Ye, Department of Mathematics and Statistics University of North Carolina University Wilmington


2018.12.04 10:00-11:00


Middle Lecture Room, Math Building


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.