Abstract
We study the influence of the multipliers ξ(n) on the angular distribution of zeroes of the Taylor series \({F_\xi }\left( z \right) = \sum\limits_{n \geqslant 0} {\xi \left( n \right)} \frac{{{z^n}}}{{n!}}\).
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.
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To Alex Eremenko on the occasion of his birthday
A. Nishry was supported by U.S. National Science Foundation Grant DMS-1128155.
A. Nishry and M. Sodin were supported by Grant No. 166/11 of the Israel Science Foundation of the Israel Academy of Sciences and Humanities.
M. Sodin was supported by Grant No. 2012037 of the United States–Israel Binational Science Foundation.
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Borichev, A., Nishry, A. & Sodin, M. Entire functions of exponential type represented by pseudo-random and random Taylor series. JAMA 133, 361–396 (2017). https://doi.org/10.1007/s11854-017-0037-0
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DOI: https://doi.org/10.1007/s11854-017-0037-0