Random Matrix Theory and Refined Large Deviations of the Riemann Zeta Function

The analogy between the Riemann zeta function and the topic of random matrix theory was first established by Keating and Snaith [35]. Following this, many mathematicians have attempted to answer number-theoretic questions using random matrix theory. One such question is the maximum of the Riemann zeta function up to a height T along the critical line, and Farmer, Gonek, Hughes established a conjecture for this maximum [21].

Our work builds upon these ideas and goes further in that we compute refined large deviations results for the characteristic polynomial of a random CbetaE matrix, generalising the CUE results of Farmer, Gonek and Hughes. We then apply these ideas to the Riemann zeta function, where we utilise a Hybrid Euler-Hadamard result of Gonek, Hughes and Keating [26] and compute refined large deviations results for each of these models separately.

Our results are consistent with the original works cited throughout this thesis, however there are some difference between the results of the two models, which we attempt to clarify here. Numerical results are presented to support (where possible) the results in this thesis.