  Following Selberg it is known that uniformly for V << (logloglog T)^{1/2 -
\epsilon} the measure of those t \in [T;2T] for which log |\zeta(1/2 + it)| >
V*((1/2)loglog T)^{1/2} is approximately T times the probability that a
standard Gaussian random variable takes on values greater than V. We extend the
range of V to V << (loglog T)^{1/10 - \epsilon}. We also speculate on the size
of the largest V for which this normal approximation can hold and on the
correct approximation beyond that point.
