  For $V\sim \alpha \log\log T$ with $0<\alpha<2$, we prove \[
\frac{1}{T}\text{meas}\{t\in [T,2T]: \log|\zeta(1/2+ {\rm i} t)|>V\}\ll
\frac{1}{\sqrt{\log\log T}} e^{-V^2/\log\log T}. \] This improves prior results
of Soundararajan and of Harper on the large deviations of Selberg's Central
Limit Theorem in that range, without the use of the Riemann hypothesis. The
result implies the sharp upper bound for the fractional moments of the Riemann
zeta function proved by Heap, Radziwi{\l}{\l} and Soundararajan. It also shows
a new upper bound for the maximum of the zeta function on short intervals of
length $(\log T)^\theta$, $0<\theta <3$, that is expected to be sharp for
$\theta > 0$. Finally, it yields a sharp upper bound (to order one) for the
moments on short intervals, below and above the freezing transition. The proof
is an adaptation of the recursive scheme introduced by Bourgade,
Radziwi{\l}{\l} and one of the authors to prove fine asymptotics for the
maximum on intervals of length $1$.
