Fluctuations for zeros of Gaussian Taylor series

Avner Kiro*, Alon Nishry

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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).

Original languageEnglish
Pages (from-to)1172-1203
Number of pages32
JournalJournal of the London Mathematical Society
Issue number3
StatePublished - Oct 2021


  • 30B20 (primary)
  • 30C15 (secondary)


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