solidot新版网站常见问题,请点击这里查看。
消息
本文已被查看79次
Large deviations and continuity estimates for the derivative of a random model of $\log |\zeta|$ on the critical line. (arXiv:1807.04860v1 [math.PR])
来源于:arXiv
In this paper, we study the random field \begin{equation*}
X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}},
\quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an
i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$.
Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good model for $(\log
|\zeta(\frac{1}{2} + i (T + h))|, \, h\in [0,1])$ when $T$ is large, if we
assume the Riemann hypothesis. The asymptotics of the maximum were found in
Arguin, Belius & Harper (2017) up to the second order, but the tightness of the
recentered maximum is still an open problem. As a first step, we provide large
deviation estimates and continuity estimates for the field's derivative
$X'(h)$. The main result shows that, with probability arbitrarily close to $1$,
\begin{equation*}
\max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1), \end{equation*}
where $\mathcal{S}$ a discrete set containing $O(\log T \sqrt{\log \log T} 查看全文>>