Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2021-12-20 DOI:10.1080/10586458.2021.2011806

E. Amzallag, L.-P. Arguin, E. Bailey, K. Huib, R. Rao

引用次数: 0

Abstract

Selberg’s central limit theorem states that the values of Abstract Image $log | ζ (1 / 2 + i τ) |$ , where τ is a uniform random variable on $[T, 2 T]$ , are asymptotically distributed like a Gaussian random variable of mean 0 and standard deviation $\sqrt{\frac{1}{2} log log T}$ . It was conjectured by Radziwiłł that this distribution breaks down for values of order Abstract Image $log log T$ , where a multiplicative correction C_k would be present at level $k log log T$ , k > 0. This constant should be the same as the one conjectured by Keating and Snaith for the leading asymptotic of the $2 k^{t h}$ moment of ζ. In this paper, we provide numerical and theoretical evidence for this conjecture. We propose that this correction has a significant effect on the distribution of the maximum of Abstract Image $log | ζ |$ in intervals of size ${(log T)}^{θ}, θ > 0$ . The precision of the prediction enables the numerical detection of C_k even for low T’s of order $T = 10^{8}$ . A similar correction appears in the large deviations of the Keating–Snaith central limit theorem for the logarithm of the characteristic polynomial of a random unitary matrix, as first proved by Féray, Méliot and Nikeghbali.

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Selberg中心极限定理大偏差的随机矩阵修正证据

【摘要】selberg中心极限定理指出，当τ为[T,2T]上的均匀随机变量时，log |ζ(1/2+iτ)|的值近似于均值为0，标准差为12log log T的高斯随机变量，渐近分布。根据Radziwiłł的推测，这种分布在log log T阶的情况下会被打破，在k log log T, k > 0的水平上会出现一个乘法修正Ck。这个常数应该与Keating和Snaith对ζ的第20阶矩的首渐近所推测的常数相同。在本文中，我们为这一猜想提供了数值和理论证据。我们认为这种修正对log |ζ|在大小为(log T)θ， θ>0的区间内的最大值的分布有显著的影响。预测的精度使得即使在T=108阶的低T下也能对Ck进行数值检测。一个类似的修正出现在随机酉矩阵特征多项式对数的Keating-Snaith中心极限定理的大偏差中，这是由fsamray, msamliot和Nikeghbali首先证明的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.