CMSA Probability Seminar: Large deviations of Selberg’s central limit theorem

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Selberg’s CLT concerns the typical behaviour of the Riemann zeta function and shows that the random variable $\Re \log \zeta(1/2 + i t)$, for a uniformly drawn $t$, behaves as a Gaussian random variable with a particular variance.  It is natural to investigate how far into the tails this Gaussianity persists, which is the topic of this work. There are also very close connections to similar problems in circular unitary ensemble characteristic polynomials.  It transpires that a `multiscale scheme’ can be applied to both situations to understand these questions of large deviations, as well as certain maxima and moments. In this talk I will focus more on the techniques we apply to approach this problem and I will assume no number theoretic knowledge. This is joint work with Louis-Pierre Arguin.