Dissertations, Theses, and Capstone Projects

Date of Degree

9-2025

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy

Program

Mathematics

Advisor

Louis-Pierre Arguin

Committee Members

Jay Rosen

Tobias Johnson

Subject Categories

Analysis | Number Theory

Keywords

Extreme Value Theory, Riemann zeta function, Probabilistic Number Theory

Abstract

We study the large values of a random model of the Riemann zeta function over short intervals. The extreme value statistics depend on the interval size: the log-correlated regime governs intervals of order one while the i.i.d. regime emerges over longer intervals. The main focus is to describe the transition between these two well-understood regimes as the interval varies in length. This thesis shows that there is an intermediate regime where the behavior of the zeta model’s maxima cannot be entirely captured by either extreme— i.i.d. or fully log-correlated. This suggests that the Riemann zeta function exhibits correlations around its extreme values and does not behave like a collection of independent random variables.

To study the hybrid statistics, we analyze the model over intervals that are parame- terized by α ∈(0,1). The main result gives matching upper and lower tail bounds for the distribution of the maximum, and describes the intermediate regime. The tail bounds in- terpolate between that of the log-correlated and i.i.d. regimes. This result refines the work of Arguin, Dubach, and Hartung, who identified the interpolating subleading term of the maximum but not the precise tail behavior or order-one fluctuations.

Further, we study the moments of the random model over the same parameterized inter- vals and obtain a new, α-dependent normalization. This analysis reveals a regime transition at α= 1/2. The inspiration for this thesis is the Fyodorov-Hiary-Keating conjecture, which pioneered the framework for studying the local maxima of the Riemann zeta function.

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