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

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".