  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).
