Moments of the Riemann zeta function on short intervals of the critical line-猫眼课题宝

DOI：

10.1214/21-aop1524

发表时间：

2019-01

期刊：

The Annals of Probability

影响因子：

通讯作者：

中科院分区：

其他

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作者：

L. Arguin;Frédéric Ouimet;Maksym Radziwill

我们证明，当$T\to\infty$时，对于所有$t\in[T,2T]$，除了一个测度为$o(T)$的集合外， \[ \int_{-(\log T)^{\theta}}^{(\log T)^{\theta}}|\zeta(\frac{1}{2}+it + ih)|^{\beta}dh = (\log T)^{f_{\theta}(\beta)+o(1)}, \] 其中$f_{\theta}(\beta)$是某个明确的指数，$\theta > - 1$且$\beta > 0$。这证明了费奥多罗夫和基廷（2014年）一个猜想的扩展版本。特别地，它表明对于所有$\theta > -1$，矩在一个临界指数$\beta_c(\theta)$处呈现相变，在该临界指数之下$f_{\theta}(\beta)$是二次的，在其之上$f_{\theta}(\beta)$是线性的。指数$f_{\theta}$的形式在介观区间（$-1 <\theta\leqslant0$）和宏观区间（$\theta > 0$）之间也有所不同，这种现象源于$\zeta$相关性的近似树结构。我们还证明，对于所有$t\in[T,2T]$，除了一个测度为$o(T)$的集合外， \[ \max_{|h|\leq(\log T)^{\theta}}|\zeta(\frac{1}{2}+it + ih)| = (\log T)^{m(\theta)+o(1)}, \] 其中$m(\theta)$是某个明确的量。这推广了纳朱德尔（2018年）以及阿尔甘等人（2018年）在$\theta = 0$时的早期结果。这些证明是无条件的，但当$\theta > 3$时的上界证明除外，此时假定了黎曼假设。

We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of measure $\mathrm{o}(T)$, $$ \int_{-(\log T)^{\theta}}^{(\log T)^{\theta}} |\zeta(\tfrac 12 + \mathrm{i} t + \mathrm{i} h)|^{\beta} \mathrm{d} h = (\log T)^{f_{\theta}(\beta) + \mathrm{o}(1)}, $$ for some explicit exponent $f_{\theta}(\beta)$, where $\theta > -1$ and $\beta > 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it show that, for all $\theta > -1$, the moments exhibit a phase transition at a critical exponent $\beta_c(\theta)$, below which $f_\theta(\beta)$ is quadratic and above which $f_\theta(\beta)$ is linear. The form of the exponent $f_\theta$ also differs between mesoscopic intervals ($-1 0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $t\in [T,2T]$ outside a set of measure $\mathrm{o}(T)$, $$ \max_{|h| \leq (\log T)^{\theta}} |\zeta(\tfrac{1}{2} + \mathrm{i} t + \mathrm{i} h)| = (\log T)^{m(\theta) + \mathrm{o}(1)}, $$ for some explicit $m(\theta)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2018) for $\theta = 0$. The proofs are unconditional, except for the upper bounds when $\theta > 3$, where the Riemann hypothesis is assumed.

Moments of the Riemann zeta function on short intervals of the critical line

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