Emma Bailey

Large deviations of random matrix and number theoretic central limit theorems

Central limit theorems inform us about the typical size of a random variable. For example, we know that for a random unitary matrix $U$, its characteristic polynomial is typically of size roughly $\exp(\sqrt{ 1/2 \log N} )$ thanks to the central limit theorem of Keating and Snaith. One might be interested in atypical behaviour, for example what is the likelihood of the characteristic polynomial being abnormally large? This would in turn provide a conjecture for the probability of large values of the Riemann zeta function. Joint with Louis-Pierre Arguin we prove (presumably) sharp upper bounds for large deviations of zeta which match the corresponding known random matrix result. The talk will assume no number theoretic knowledge (though local experts are welcome!).

Talk recording: https://mediasite.bris.ac.uk/Mediasite/Play/b80dc304a8314b58a52736d7be6862f31d