High points of a random model of the Riemann-zeta function and Gaussian multiplicative chaos (Q2157324)

Let \(\mathcal{P}\) be the set of all prime numbers and \((\theta_p)_{p \in \mathcal{P}}\) be i.i.d. uniform \([0, 2 \pi]\) random variables. For \(N \in \mathbb{N}\), \textit{A. J. Harper} [``A note on the maximum of the Riemann zeta function, and log-correlated random variables'', Preprint, \url{arXiv:1304.0677}] proposed the following model for large values of the Riemann-Zeta function on a typical interval of length 1 of the critical line: \[ X_N(x) = \sum_{p\in \mathcal{P} \cup [0,N]} \frac{1}{\sqrt{p}} (\cos (x \ln p) \cos (\theta_P) + sin (x \ln p) \sin (\theta_p)), x \in [0,1]. \] In this article, the authors are interested in the values of this process of the order of \(\frac{\alpha}{2} \ln \ln N\) with \(\alpha < 2\). If the Lebesgue measure of \(\alpha\)-high points \(W_{\alpha,N} = \mathrm{Leb} \{x \in [0,1]: X_N(x) > \frac{\alpha}{2} \ln \ln N\}, \) the main result of the article is: for any \(0 < \alpha < 2, \frac{W_{\alpha, N}}{\mathbb{E}(W_{\alpha, N})} \rightarrow M_{\alpha}\) in probability as \(N \rightarrow \infty\), where \(\mathbb{E}\) denotes the expectation with respect to \(\theta_p\)'s, \(M_{\alpha} = \lim_{N \rightarrow \infty} \int_{0}^{1} M_{\alpha, N}(dx)\) a.s., \(M_{\alpha, N} = \frac{e^{\alpha X_{N}(x)}}{\mathbb{E}(e^{\alpha X_N(x)})} dx\), the random measure. For the Riemann-Zeta function \(\zeta\), the authors conjecture the following: If \(\tau\) is uniform random variable on \([T, 2T]\), \(W_{\alpha, T} = \mathrm{Leb}\{h \in [0,1]: \ln \mid \zeta(1/2 + i (\tau + h))\mid > \frac{\alpha}{2}\ln \ln T\}\), then, for \(\alpha < 2\), \[ \lim_{T \rightarrow \infty} \frac{W_{\alpha, T}}{\mathbb{E}(W_{\alpha, T})} = \lim_{T \rightarrow \infty} \frac{\int_{0}^{1} \mid \zeta(1/2 + i(\tau + h))\mid^{\alpha}dh}{\mathbb{E}(\mid \zeta(1/2 + i \tau)\mid^{\alpha})} \] in probability. The proof of the main result is based on a first and a second moment estimate and a global strategy for branching Brownian motion.