Assuming the Riemann Hypothesis, we will present a proof that for $k>0$, $$\frac{1}{T}{\rm meas}\Big\{t\in [T,2T]:|\zeta(1/2+{\rm i} t)|>(\log T)^k\Big\}\leq C_k \frac{(\log T)^{-k^2}}{\sqrt{\log\log T}},$$ where $C_k=e^{k^A}$ for some absolute constant $A>0$. This implies that the $2k$-moments of $|\zeta|$ are bounded above by $C_k(\log T)^{k^2}$, improving the bound $e^{e^{O(k)}}(\log T)^{k^2}$ of Harper. The proof relies on the recursive scheme of a prior work with Bourgade and Radziwiłł, and combines ideas of Soundararajan and Harper. We will discuss the connections with the Keating-Snaith Conjecture from Random Matrix Theory for the optimal $C_k$. This is joint work with Emma Bailey and Asher Roberts.
