On convergence of points to limiting processes, with an application to zeta zeros (Q6617243)

In this paper under review, the authors discuss first the non trivial zeros \(\rho_n=\sigma_n+i\gamma_n\) of the Riemann zeta function \(\zeta(s)\), with \(\sigma_n \in (0,1)\), \(\gamma_n\in \mathbb{R}\) and \(\gamma_n\leq \gamma_{n+1}\). One possible rescaling of \(\gamma_n\) is \(\tilde{\gamma}_n:=\frac{1}{\pi}\upsilon(\gamma_n)\), where \(\upsilon(t):=\Im(\log \Gamma(1/4+i t/2))-\frac{t}{2}\log \pi\).\N\NThe main result of the paper is to show that the following notions are equivalent:\N\begin{itemize}\N\item[(i)] The GUE Hypothesis (that is, for \(t \in [T, 2T ]\) chosen uniformly at random, the point processes \(\{\frac{\log T}{2\pi} (\gamma_n-t)\}\) tend in correlation to the sine-kernel process as \(T \to \infty\)).\N\item[(ii)] The point processes \(\{\tilde{\gamma}_n-t\}\), for \(t \in (0, T ]\) chosen uniformly at random, tend in correlation to the sine-kernel process as \(T\to \infty\).\N\item[(iii)] The point processes \(\{\tilde{\gamma}_n -t\}\), for \(t \in (0, T ]\) chosen uniformly at random, tend in distribution to the sine-kernel process as \(T\to \infty\).\N\item[(iv)] The joint distribution of spacings between zeros \({\tilde{\gamma}_n}\) tends to the joint distribution of spacings between points in the sine-kernel process.\N\end{itemize}