Large Deviation Estimates of Selberg's Central Limit Theorem and Applications

Louis-Pierre Arguin, Emma Bailey

For $V\sim \alpha \log \log T$ with $0<\alpha <2$, we prove \[ \frac{1}{T}\text{meas}\{t\in [T,2T]: \log|ζ(1/2+ {\rm i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^2/\log\log T}. \] This impro…