论文标题
Fyodorov-Hiary-Keating猜想。我
The Fyodorov-Hiary-Keating Conjecture. I
论文作者
论文摘要
通过类似于随机矩阵的猜想，fyodorov-hiary-keating和fyodorov-keating提出的精确渐近性，以在临界线上的典型短间隔中最大程度地提出了riemann zeta功能。在本文中，我们以强烈的形式解决了他们猜想的上限。更确切地说，我们证明了这些$ t \ leq t \ leq 2t $的度量 $$ \ max_ {| h | \ leq 1} |ζ（1/2 + i t + i h）| > e^y \ frac {\ log t} {（\ log \ log t）^{3/4}} $$由$ cy e^{ - 2y} $统一在$ y \ geq 1 $中均匀。预计这将是$ y = o（\ sqrt {\ log \ log t}）$的最佳选择。该上限比随机矩阵中已知的限制更加清晰，因为它以$ y $的价格（均匀）衰减率。在随后的论文中，我们将获得匹配的下限。
By analogy with conjectures for random matrices, Fyodorov-Hiary-Keating and Fyodorov-Keating proposed precise asymptotics for the maximum of the Riemann zeta function in a typical short interval on the critical line. In this paper, we settle the upper bound part of their conjecture in a strong form. More precisely, we show that the measure of those $T \leq t \leq 2T$ for which $$ \max_{|h| \leq 1} |ζ(1/2 + i t + i h)| > e^y \frac{\log T }{(\log\log T)^{3/4}}$$ is bounded by $Cy e^{-2y}$ uniformly in $y \geq 1$. This is expected to be optimal for $y= O(\sqrt{\log\log T})$. This upper bound is sharper than what is known in the context of random matrices, since it gives (uniform) decay rates in $y$. In a subsequent paper we will obtain matching lower bounds.
