Large deviations in Selberg's central limit theorem

Following Selberg it is known that uniformly for V << (logloglog T)^{1/2 - ε} the measure of those t \in [T;2T] for which log |ζ(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 - ε}. 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.

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