We improve the current bounds for an inequality of Erdős and Turán from 1950 related to the discrepancy of angular equidistribution of the zeros of a given polynomial. Building upon a recent work of Soundararajan, we establish a novel connection between this inequality and an extremal problem in Fourier analysis involving the maxima of Hilbert transforms, for which we provide a complete solution. Prior to Soundararajan (2019), refinements of the discrepancy inequality of Erdős and Turán had been obtained by Ganelius (1954) and Mignotte (1992).

中文翻译：

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希尔伯特变换和多项式零点的等分布

我们改进了 1950 年以来 Erdős 和 Turán 不等式的当前边界，该不等式与给定多项式的零点的角度等分布的差异有关。基于 Soundararajan 最近的工作，我们在这个不等式与傅立叶分析中涉及希尔伯特变换最大值的极值问题之间建立了一种新的联系，为此我们提供了一个完整的解决方案。在 Soundararajan (2019) 之前，Ganelius (1954) 和 Mignotte (1992) 已经对 Erdős 和 Turán 的差异不等式进行了改进。