论文标题
塞尔伯格中央限制定理和应用的大偏差估计值
Large Deviation Estimates of Selberg's Central Limit Theorem and Applications
论文作者
论文摘要
对于$ v \simα\ log \ log t $带有$ 0 <α<2 $，我们证明\ [\ frac {1} {t} {t} \ text {scal} \ { \ frac {1} {\ sqrt {\ log \ log t}} e^{ - v^2/\ log \ log \ log t}。 \]这在不使用Riemann假设的情况下，这改善了Soundararajan和Harper在Selberg的中心极限定理的巨大偏差上的先前结果。结果意味着在Riemann Zeta功能的分数时刻，由Heap，Radziwiłł和Soundararajan证明了尖锐的上限。它还显示了一个新的上限，以短期$（\ log t）^θ$，$ 0 <θ<3 $的短间隔为Zeta函数的最大界限，预计将对$θ> 0 $进行清晰。最后，它在短时间间隔（下方）下方和上方的矩矩时产生尖锐的上限（订购一个）。证明是对Bourgade，Radziwiłł和一位作者提出的递归计划的改编，以证明最大的渐近造型，以$ 1 $ $ 1 $的长度间隔。
For $V\sim α\log\log T$ with $0<α<2$, we prove \[ \frac{1}{T}\text{meas}\{t\in [T,2T]: \log|ζ(1/2+ {\rm i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^2/\log\log T}. \] This improves prior results of Soundararajan and of Harper on the large deviations of Selberg's Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwiłł and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $(\log T)^θ$, $0<θ<3$, that is expected to be sharp for $θ> 0$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwiłł and one of the authors to prove fine asymptotics for the maximum on intervals of length $1$.
