We study fluctuations in the number of zeros of random analytic functions given by a Taylor series whose coefficients are independent complex Gaussians. When the functions are entire, we find sharp bounds for the asymptotic growth rate of the variance of the number of zeros in large disks centered at the origin. To obtain a result that holds under no assumptions on the variance of the Taylor coefficients, we employ the Wiman–Valiron theory. We demonstrate the sharpness of our bounds by studying well-behaved covariance kernels, which we call admissible (after Hayman).

中文翻译：

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高斯泰勒级数零点的波动

我们研究由系数为独立复高斯分布的泰勒级数给出的随机解析函数的零点数的波动。当函数是完整的时，我们发现以原点为中心的大圆盘中零数量的方差的渐近增长率有明显的界限。为了获得在泰勒系数方差没有假设的情况下成立的结果，我们采用了 Wiman-Valiron 理论。我们通过研究表现良好的协方差核来证明我们界限的尖锐性，我们称之为可接受的（在海曼之后）。