Moments of the Riemannn zeta function and log-correlated random variables - Curran, MJ (2025)

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Recent developments in random matrix theory and the theory of log-correlated random variables have led to several interesting conjectures in analytic number theory. This conjectures primarily concern the asymptotic behavior of several quantities related to the moments of the zeta function. In this dissertation, we prove estimates of the conjectured order of magnitude for several of these quantities. First we will give upper bounds for the joint moments of the 2kth power of the zeta function with the 2hth power of its logarithmic derivative when k ∈ [1, 2] and h ∈ [0, 1]. Then in joint work with Andr´e Heycock, we extend these upper bounds to all 0 ≤ h ≤ k ≤ 2 unconditionally and for all 0 ≤ h ≤ k assuming the Riemann hypothesis. We also prove unconditional lower bounds for all 0 ≤ h ≤ k, demonstrating that our upper bounds are of the correct order of magnitude.