The "pits effect" for entire functions of exponential type and the Wiener spectrum
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| Format: | Preprint |
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2019
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| _version_ | 1866917453377830912 |
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| author | Benatar, Jacques Borichev, Alexander Sodin, Mikhail |
| author_facet | Benatar, Jacques Borichev, Alexander Sodin, Mikhail |
| contents | Given a sequence $ξ\colon \mathbb Z_+ \to \mathbb C$, we find a simple spectral condition which guarantees the angular equidistribution of the zeroes of the Taylor series \[ F_ξ(z) = \sum_{n\ge 0} ξ(n) \frac{z^n}{n!}\,. \] This condition yields practically all known instances of random and pseudo-random sequences $ξ$ with this property (due to Nassif, Littlewood, Chen-Littlewood, Levin, Eremenko-Ostrovskii, Kabluchko-Zaporozhets, Borichev-Nishry-Sodin), and provides several new ones. Among them are Besicovitch almost periodic sequences and multiplicative random sequences. It also conditionally yields that the Möbius function $μ$ has this property assuming "the binary Chowla conjecture". |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1908_09161 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The "pits effect" for entire functions of exponential type and the Wiener spectrum Benatar, Jacques Borichev, Alexander Sodin, Mikhail Probability Complex Variables Given a sequence $ξ\colon \mathbb Z_+ \to \mathbb C$, we find a simple spectral condition which guarantees the angular equidistribution of the zeroes of the Taylor series \[ F_ξ(z) = \sum_{n\ge 0} ξ(n) \frac{z^n}{n!}\,. \] This condition yields practically all known instances of random and pseudo-random sequences $ξ$ with this property (due to Nassif, Littlewood, Chen-Littlewood, Levin, Eremenko-Ostrovskii, Kabluchko-Zaporozhets, Borichev-Nishry-Sodin), and provides several new ones. Among them are Besicovitch almost periodic sequences and multiplicative random sequences. It also conditionally yields that the Möbius function $μ$ has this property assuming "the binary Chowla conjecture". |
| title | The "pits effect" for entire functions of exponential type and the Wiener spectrum |
| topic | Probability Complex Variables |
| url | https://arxiv.org/abs/1908.09161 |