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 |$\tfrac{1}{2}\mathbb{Z}$|⁠, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known about zeros of the zeta function.

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更高的相关性和替代假设

替代假设（AH）涉及一种假设性的，不太可能的图片，即关于Riemann zeta函数的零点如何隔开的，这是我们想排除的。在替代假设中，非平凡零之间的重新归一化距离应该始终位于半个整数。已知替代假设与关于zeta零的对相关函数的已知相容。我们想知道关于零的更高相关函数的当前已知信息是否足以排除“替代假设”，并通过构造一个明确的反例点过程来证明它不是。陶最近使用不同的方法独立获得了相似的结果。| $ \ tfrac {1} {2} \ mathbb {Z} $ |⁠，它满足替代假设间隔，但模拟了有关zeta函数零的当前已知局部统计信息。