Entire functions of exponential type represented by pseudo-random and random Taylor series

We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series Fξ(z)=∑nn)znn!. We show that the distribution of zeroes of Fξ is governed by certain autocorrelations of the sequence ξ. Using this guiding principle, we consider several examples of random and pseudo-random sequences ξ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product, we show that if ξ is a stationary random integer-valued sequence, then either it is periodic, or its spectral measure has no gaps in its support. The same conclusion is true if ξ is a complex-valued stationary ergodic sequence that takes values in a uniformly discrete set.