Moments of the Riemann zeta function on short intervals of the critical line. (English) Zbl 1497.11194
Summary: We show that as \(T\to\infty\), for all \(t\in[T,2T]\) outside of a set of measure \(\mathrm{o}(T)\),
\[
\int_{-\log^\theta T}^{\log^\theta T}\left |\zeta\left(\frac{1}{2}+\mathrm{i}t+\mathrm{i}h\right)\right|^\beta \mathrm{d}h=(\log T)^{f_\theta(\beta)+\mathrm{o}(1)},
\]
for some explicit exponent \(f_{\theta}(\beta)\), where \(\theta > -1\) and \(\beta > 0\). This proves an extended version of a conjecture of Y. V. Fyodorov and J. P. Keating [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 372, No. 2007, Article ID 20120503, 32 p. (2014; Zbl 1330.82028)]. In particular, it shows 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 \((-1 < \theta < 0)\) and macroscopic intervals \((\theta > 0)\), 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 \(o(T)\),
\[
\max\limits_{|h|\leq\log^\theta T}\left|\left(\frac{1}{2}+\mathrm{i}t+\mathrm{i}h\right)\right|=(\log t)^{m(\theta)+\mathrm{o}(1)},
\]
for some explicit \[m(\theta )\]. This generalizes earlier results of J. Najnudel [Probab. Theory Relat. Fields 172, No. 1–2, 387–452 (2018; Zbl 1442.11125)] and L.-P. Arguin et al. [Commun. Pure Appl. Math. 72, No. 3, 500–535 (2019; Zbl 1443.11161)] for \[\theta =0\]. The proofs are unconditional, except for the upper bounds when \(\theta > 3\), where the Riemann hypothesis is assumed.
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 60G70 | Extreme value theory; extremal stochastic processes |
| 60F10 | Large deviations |
| 60G60 | Random fields |
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