Zero distribution of power series and binary correlation of coefficients

TL;DR

This paper investigates the zero distribution of power series with coefficients influenced by binary correlations, establishing conditions for equidistribution relative to a radial measure.

Contribution

It introduces a framework connecting binary correlations of coefficients to the equidistribution of zeros in power series with infinite radius of convergence.

Findings

Zeros are equidistributed with respect to a radial measure under certain conditions.

The approach applies to various sequences including IID, quadratic exponential, and automatic sequences.

Abstract

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.