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 $$ max_{|h| leq 1} |zeta(1/2 + i t + i h)| > e^y frac{log T }{(loglog T)^{3/4}}$$ is bounded by $Cy e^{-2y}$ uniformly in $y geq 1$. This is expected to be optimal for $y= O(sqrt{loglog T})$. 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.
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