  By analogy with conjectures for random matrices, Fyodorov-Hiary-Keating and
Fyodorov-Keating proposed precise asymptotics for the maximum of the Riemann
zeta function in a typical short interval on the critical line. In this paper,
we settle the upper bound part of their conjecture in a strong form. More
precisely, we show that the measure of those $T \leq t \leq 2T$ for which
  $$
  \max_{|h| \leq 1} |\zeta(1/2 + i t + i h)| > e^y \frac{\log T }{(\log\log
T)^{3/4}}$$ is bounded by $Cy e^{-2y}$ uniformly in $y \geq 1$. This is
expected to be optimal for $y= O(\sqrt{\log\log T})$. This upper bound is
sharper than what is known in the context of random matrices, since it gives
(uniform) decay rates in $y$. In a subsequent paper we will obtain matching
lower bounds.
