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 $F_ξ(z) = \sum_{n\ge 0} ξ(n) \frac{z^n}{n!}\,.$ We show that the distribution of zeroes of $F_ξ$ 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 from a uniformly discrete set.

No reviews yet.

No comments yet.