zeta零点的介观波动

摘要：

我们证明了Selberg的中心极限定理的多维扩展$\logζ$，其中出现非平凡的相关性。特别是，这回答了一个问题的Coram和Diaconis关于介观波动的零点的黎曼ζ函数。在随机矩阵的背景下，从酉群给出了类似的结果。这不仅示出了矩阵的尺寸与临界线上的高度之间的对应关系$n \leftrightarrow \log t$，而且还示出了在局部尺度上对于与临界轴或单位圆的小偏差的对应关系。

英文摘要：

We prove a multidimensional extension of Selberg's central limit theorem for $\logζ$, in which non-trivial correlations appear. In particular, this answers a question by Coram and Diaconis about the mesoscopic fluctuations of the zeros of the Riemann zeta function. Similar results are given in the context of random matrices from the unitary group. This shows the correspondence $n \leftrightarrow \log t$ not only between the dimension of the matrix and the height on the critical line, but also, in a local scale, for small deviations from the critical axis or the unit circle.

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