On the extreme values of the Riemann zeta function on random intervals of the critical line

In the present paper, we show that under the Riemann hypothesis, and for fixed h, epsilon > 0, the supremum of the real and the imaginary parts of log.(1/ 2 + i t) for t. [ UT -h, UT + h] are in the interval [(1 -epsilon) log log T, (1 + epsilon) log log T] with probability tending to 1 when T goes to infinity, U being a uniform random variable in [ 0, 1]. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating, which has recently been intensively studied in the setting of random matrices. We also unconditionally show that the supremum of epsilon log.(1/ 2 + i t) is at most log log T + g(T) with probability tending to 1, g being any function tending to infinity at infinity.