幂级数零分布与系数二元相关性
Zero distribution of power series and binary correlation of coefficients
AI总结：
研究幂级数零点分布，系数形式为ξ(n)a(n)，通过二元相关性和无谱间隙的乘子序列，证明零点均匀分布于由a(n)定义的径向测度。
摘要：
我们研究了无限半径收敛幂级数零点分布。系数形式为ξ(n)a(n)，其中a是正数光滑序列，ξ是具有二元相关性和无谱间隙的复值乘子序列。在a的光滑性和ξ的二元相关性假设下，证明幂级数的零点相对于由a(n)定义的径向测度均匀分布。我们应用此方法到几种ξ序列的例子：(i) 独立同分布序列，(ii) 有迪菲安特α的e(αn²)序列，(iii) 随机乘法序列，(iv) 吉莱-鲁丁-沙普尔序列，(v) 无平方因子整数的指示函数，(vi) 托-莫尔斯序列。
英文摘要：
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 $e(αn^2)$ 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.
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