In this talk, we study the maximum of the real and the imaginary part of the logarithm of the Riemann zeta function on random intervals of the critical line. We sketch a proof of the fact that if these intervals are centered at 1/2 + i UT, U being uniformly distributed on [0,1], then the maximum, divided by log log T, tends to 1 in probability. This proves a weak version of a conjecture by Fyodorov, Hiary and Keating.