  We argue that the freezing transition scenario, previously conjectured to
occur in the statistical mechanics of 1/f-noise random energy models, governs,
after reinterpretation, the value distribution of the maximum of the modulus of
the characteristic polynomials p_N(\theta) of large N\times N random unitary
(CUE) matrices; i.e. the extreme value statistics of p_N(\theta) when N
\rightarrow\infty. In addition, we argue that it leads to multifractal-like
behaviour in the total length \mu_N(x) of the intervals in which
|p_N(\theta)|>N^x, x>0, in the same limit. We speculate that our results extend
to the large values taken by the Riemann zeta-function \zeta(s) over stretches
of the critical line s=1/2+it of given constant length, and present the results
of numerical computations of the large values of \zeta(1/2+it). Our main
purpose is to draw attention to the unexpected connections between these
different extreme value problems.
