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

TL;DR

This paper establishes a spectral condition ensuring the zeroes of certain exponential type entire functions are angularly equidistributed, unifying known results and introducing new classes of sequences with this property.

Contribution

It provides a simple spectral criterion that guarantees zeroes' equidistribution for entire functions generated by various sequences, including new classes like Besicovitch almost periodic and multiplicative random sequences.

Findings

The spectral condition covers all known cases of random and pseudo-random sequences.

It introduces new classes of sequences, such as Besicovitch almost periodic sequences.

Conditional results relate the M"obius function to zeroes' distribution under the binary Chowla conjecture.

Abstract

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".