ZERO DISTRIBUTION OF POWER SERIES AND BINARY CORRELATION OF COEFFICIENTS

We study the distribution of zeroes of power series with infinite radius of convergence. The coefficients of the series have the form ξ(n)a(n), where a is a smooth sequence of positive numbers, and ξ 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 ξ, 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 ξ: (i) IID sequences, (ii) sequences e(αn²) with Diophantine α, (iii) random multiplicative sequences, (iv) the Golay–Rudin–Shapiro sequence, (v) the indicator function of the square-free integers, (vi) the Thue–Morse sequence.