Higher correlations and the alternative hypothesis
Higher correlations and the alternative hypothesis
复制标题
更高的相关性和替代假设
DOI：
10.1093/qmathj/haz.043
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Rodgers, Brad
中科院分区：
文献类型：
--
作者：
Lagarias, Jeffrey C.;Rodgers, Brad
另类假设 (AH) 涉及黎曼 zeta 函数的零点如何间隔的假设且不太可能的图像，人们希望排除这一点。在备择假设中，非平凡零点之间的重整化距离应该始终位于半整数。众所周知，备择假设与 zeta 零点配对相关函数的已知情况是兼容的。我们询问当前已知的零点较高相关函数是否足以排除替代假设，并通过构建明确的反例点过程来证明事实并非如此。最近，Tao 使用略有不同的方法独立获得了类似的结果。我们还将遍历定理应用于该点过程，以表明存在确定性的点集合，这些点满足备择假设间距，但模仿了当前已知的关于 zeta 函数零点的局部统计数据。
The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.