Large deviations in Selberg's central limit theorem
classification
🧮 math.NT
math.PR
keywords
approximationepsilonloglogselbergapproximatelybeyondcentralcorrect
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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.
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Forward citations
Cited by 1 Pith paper
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Conditional Upper Bounds for Large Deviations and Moments of the Riemann Zeta Function
Under RH, the measure of t in [T,2T] with |zeta(1/2+it)| > (log T)^k is <= C_k (log T)^{-k^2}/sqrt(log log T) with C_k=exp(e^{ck}), implying 2k-moment bounds C_k (log T)^{k^2}.
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