论文标题
Jensen多项式不是证明Riemann假设的合理途径
Jensen polynomials are not a plausible route to proving the Riemann Hypothesis
论文作者
论文摘要
关于Riemann XI功能的Jensen多项式及其衍生物的最新工作发现了与Hermite多项式的联系。已经提出这些结果为Riemann假设提供了证据，此外，人们已经提出，这些结果揭示了Zeta功能的零零矩阵统计数据。我们将其放置在先前结果的背景下，并解释了为什么Hermite多项式的出现有趣且令人惊讶，并且可能代表了一种新型的普遍定律，该法律完善了M. Berry的“余弦是普遍的吸引者”原则。但是，我们发现与Riemann假设的建议联系没有理由，也没有建议与L功能零的猜想随机矩阵统计的建议联系。这些考虑因素表明，Jensen多项式以及大量相关的多项式对攻击Riemann假设无用。我们提出了一般标准，用于确定与Riemann假设的等效性是否可能有用。
Recent work on the Jensen polynomials of the Riemann xi-function and its derivatives found a connection to the Hermite polynomials. Those results have been suggested to give evidence for the Riemann Hypothesis, and furthermore it has been suggested that those results shed light on the random matrix statistics for zeros of the zeta-function. We place that work in the context of prior results, and explain why the appearance of Hermite polynomials is interesting and surprising, and may represent a new type of universal law which refines M. Berry's "cosine is a universal attractor" principle. However, we find there is no justification for the suggested connection to the Riemann Hypothesis, nor for the suggested connection to the conjectured random matrix statistics for zeros of L-functions. These considerations suggest that Jensen polynomials, as well as a large class of related polynomials, are not useful for attacking the Riemann Hypothesis. We propose general criteria for determining whether an equivalence to the Riemann Hypothesis is likely to be useful.
