The multivariate rate of convergence for Selberg's central limit theorem (Q6591599)

Let \(\zeta(s)\) be the Riemann zeta function. Let \(\tau\) be a random point distributed uniformly on \([T, 2T ]\), and let \(|h-h'|\sim (\log T)^{-\alpha}\) for \(\alpha \in (0,1)\). It is shown that for large \(T\) \N\[\Nd_\mathcal{D}\left\lbrack \left (\frac{\log |\zeta( \frac{1}{2} + i(\tau + h))|}{\sqrt{\frac{1}{2}\log\log T}}, \frac{\log |\zeta( \frac{1}{2} + i(\tau + h'))|}{\sqrt{\frac{1}{2}\log\log T}}\right), (\mathcal{Z},\mathcal{Z}')\right\rbrack\ll \frac{(\log \log \log T) ^2}{\sqrt{\log \log T}}, \N\]\Nwhere \(d_\mathcal{D}\) denotes the Dudley distance to estimate the rate of convergence and \((\mathcal{Z},\mathcal{Z}')\) is a Gaussian vector with mean 0 and covariance matrix \( \left(\begin{array}{cc} 1 & \alpha \\\N\alpha & 1\end{array}\right)\). Actually, the author quantifies the above rate of convergence in Selberg's central limit theorem for \(\log|\zeta(\frac{1}{2}+it)|\) by using the method of proof given by \textit{M. Radziwiłł} and \textit{K. Soundararajan} [Enseign. Math. (2) 63, No. 1--2, 1--19 (2017; Zbl 1432.11117)]. This technique allows to prove the theorem for the multivariate case given by \textit{P. Bourgade} [Probab. Theory Relat. Fields 148, No. 3--4, 479--500 (2010; Zbl 1250.11080)] with the same rate of convergence as in the single variable.