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

2014/09/09 by Alexander Borichev, Borichev, Alexander, Alon Nishry +3
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.CV #math.PR

paper · pdf · doi:10.48550/arxiv.1409.2736

44 pages, to appear in Journal d'Analyse mathématique

openalex publication_date 2014/09/09 · arxiv created 2016/01/08 · arxiv updated 2016/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series F_ξ(z) = ∑_{n≥ 0} ξ(n) (zⁿ)/(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.

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