Large Deviations of the Riemann zeta function on the critical line

Selbergs Central Limit Theorem asserts that the typical value of the logarithm of the zeta function at a height $T$
on the critical line is normally distributed with a standard deviation of order $(\log \log T)^{1/2}$. In this talk, we
will discuss recent works showing that these normal fluctuations persist, up to a constant, for values of the order of
the variance $\log \log T$. The results naturally relate to the works of Soundararajan and Harper on sharp upper
bounds of the $2k$-moments, as well as the work of Heap and Soundararajan on lower bounds. The connections
with Radziwi\l\l's conjecture, and in particular estimates of the Keating--Snaith constant, will also be discussed.
This is based on joint work with E. Bailey, A. Roberts, and N. Creighton.