Entire functions of exponential type represented by pseudo-random and random Taylor series |
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Authors: | Email author" target="_blank">Alexander?BorichevEmail author Alon?Nishry Mikhail?Sodin |
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Institution: | 1.Aix Marseille Université,CNRS, Centrale Marseille,Marseille,France;2.School of Mathematics,Tel Aviv University,Tel Aviv,Israel |
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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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