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

Abstract

We study the influence of the multipliers $\xi (n)$ on the angular distribution of zeroes of the Taylor series \[ F_\xi (z) = \sum_{n\ge 0} \xi (n) \frac{z^n}{n!}\,. \] We show that the distribution of zeroes of $ F_\xi $ is governed by certain autocorrelations of the sequence $ \xi $. Using this guiding principle, we consider several examples of random and pseudo-random sequences $\xi$ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product we show that if $\xi$ 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 $\xi$ is a complex-valued stationary ergodic sequence that takes values from a uniformly discrete set.