  We study the distribution of zeroes of power series with infinite radius of
convergence. The coefficients of the series have the form $\xi(n)a(n)$, where
$a$ is a smooth sequence of positive numbers, and $\xi$ is a sequence of
complex-valued multipliers having binary correlations and no gaps in the
spectrum. We show that under certain assumptions on the smoothness of the
sequence $a$ and on the binary correlations of the multipliers $\xi$, the
zeroes of the power series are equidistributed with respect to a radial measure
defined by the sequence $a$.
  We apply our approach to several examples of the sequence $\xi$: (i) IID
sequences, (ii) sequences $e(\alpha n^2)$ with Diophantine $\alpha$, (iii)
random multiplicative sequences, (iv) the Golay--Rudin--Shapiro sequence, (v)
the indicator function of the square-free integers, (vi) the Thue--Morse
sequence.
