  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".
