How extreme can the Riemann zeta function get on the critical line Re(s) = 1/2? The Lindelöf hypothesis says not very — it bounds the growth rate. But unconditional results (those not assuming the Riemann Hypothesis) have struggled to establish tight lower bounds on the extremes. We know zeta gets large; the question is exactly how large.
Arguin & Creighton (arXiv:2603.01711) establish unconditional lower bounds for both large deviations and fractional moments of |ζ(1/2 + it)|. For deviations of magnitude V ~ α log log T (where T is the height on the critical line), their lower bounds match the known upper bounds when 0 < α < 2. In this range, the question “how wild can zeta get?” has a definitive answer: exactly this wild.
The proof provides an alternative derivation of previously known moment bounds, but through the large-deviation lens. Instead of asking “what is the average of |ζ|^{2k}?” (the moment question), they ask “how likely is |ζ| to exceed a given threshold?” (the deviation question). The two questions are related by Laplace-transform–style duality, but the deviation perspective is more geometric: it describes the shape of the distribution's tail rather than its center.
For α > 2, the gap between upper and lower bounds remains open. The transition at α = 2 is not an artifact of the proof technique — it corresponds to a genuine change in the mechanism producing large values. Below α = 2, the large values come from coincidental resonances between Dirichlet polynomial terms. Above α = 2, they would require deeper structural conspiracies that current methods can't rule out or confirm.
The critical line is wilder than you might hope, and the wildness is now quantified in the regime where it matters most.