2020/12/14 by Li, Hui, Wang, Jun, Yao, Xiao +1
#30B20 #30D15 #60G99 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.07453
Let fω(z)=∑j=0∞χj(ω) aj zj be a random entire function, where χj(ω) are independent and identically distributed random variables defined on a probability space (Ω, F, μ). In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Steinhaus entire functions. Then, we prove that, for almost all functions in the family and for any constant C>1, there exist a constant r0=r0(ω) and a set E⊂ [e, ∞) of finite logarithmic measure such that, for r>r0 and r∉ E, |log M(r, f)- N(r,0, fω)|≤ (C/A)^\frac1Blog^\frac1Blog M(r,f) +loglog M(r, f), a.s. where A, B are constants, M(r, f) is the maximum modulus, and N(r, 0, f) is the weighted counting-zero function of f. As a by-product of our main results, we prove Nevanlinna's second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by a weighed counting function, rather than by two weighted counting functions in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions fω and for any ε>0, there is r0 such that, for r>r0, T(r, f) ≤ N(r,0, fω)+(\frac12+ε) log T(r, f).