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

TL;DR

This paper investigates how the multipliers in Taylor series influence the zero distribution of entire functions of exponential type, linking zero patterns to autocorrelations of the multiplier sequences, including random and pseudo-random cases.

Contribution

It establishes a connection between zero distribution and autocorrelations of multiplier sequences, answering longstanding questions and characterizing stationary sequences in terms of spectral measures.

Findings

Zero distribution governed by autocorrelations of multipliers

Random and pseudo-random sequences exhibit specific zero patterns

Stationary sequences are either periodic or have spectral measures with no gaps

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.