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Article Dans Une Revue Journal of the London Mathematical Society Année : 2021

The "pits effect" for entire functions of exponential type and the Wiener spectrum

Résumé

Given a sequence $\xi\colon \mathbb Z_+ \to \mathbb C$, we find a simple spectral condition which guarantees the angular equidistribution of the zeroes of the Taylor series \[ F_\xi (z) = \sum_{n\ge 0} \xi (n) \frac{z^n}{n!}\,. \] This condition yields practically all known instances of random and pseudo-random sequences $\xi$ 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 M\"obius function $\mu$ has this property assuming "the binary Chowla conjecture".

Dates et versions

hal-02544070 , version 1 (15-04-2020)

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Jacques Benatar, Alexander Borichev, Mikhail Sodin. The "pits effect" for entire functions of exponential type and the Wiener spectrum. Journal of the London Mathematical Society, 2021, 104 (3), pp.1433-1451. ⟨hal-02544070⟩
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