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Maximum of the Riemann zeta function on a short interval of the critical line

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abstract

We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as $T \rightarrow \infty$ for a set of $t \in [T, 2T]$ of measure $(1 - o(1)) T$, we have $$ \max_{|t-u|\leq 1}\log\left|\zeta\left(\tfrac{1}{2}+i u\right)\right|=(1 + o(1))\log\log T . $$

fields

hep-th 1

years

2026 1

verdicts

UNVERDICTED 1

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Black Holes and Random Variables

hep-th · 2026-07-02 · unverdicted · novelty 6.0

Formulates an avatar of the Fyodorov-Hiary-Keating conjecture for black hole microstate counts, implying sharp bounds on CFT primary operator interval counts and suggesting that AdS spectra exhibit extreme value statistics of Gaussian log-correlated random matrices.

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  • Black Holes and Random Variables hep-th · 2026-07-02 · unverdicted · none · ref 45 · internal anchor

    Formulates an avatar of the Fyodorov-Hiary-Keating conjecture for black hole microstate counts, implying sharp bounds on CFT primary operator interval counts and suggesting that AdS spectra exhibit extreme value statistics of Gaussian log-correlated random matrices.