Year: 2026
Type: preprint
Abstract: Assuming the Riemann Hypothesis, we study negative moments and small values of the Riemann zeta-function to the right of the critical line. For every fixed $00$, improving on previous results of Soundararajan, Balazard and de Roton, and of the first two authors. For $t$ chosen uniformly from $[T,2T]$, we also prove that \[ \min_{|h|\leq1} \Big|ζ\Big(\tfrac12+α+i(t+h)\Big)\Big| =α^{1+o(1)} \] in probability, provided that $α\ll (\log T)^{-c}$ for some fixed $c >0$, and $α\log T/\log\log T\to\infty$. This agrees with the leading order predicted by a minimum analogue of the Fyodorov--Hiary--Keating Conjecture. Further results include bounds for minima on intervals of logarithmically growing length, for negative moments in almost all short intervals, and an extension of the range of sharp negative-moment bounds for $k>1/2$. A main ingredient in the proof of some of these results is a Gaussian large-deviation upper bound for $\log|ζ(1/2+α+it)|^{-1}$, obtained by building on work of Arguin, Bailey, and Roberts.
Source: arXiv (Cornell University)
Institutions: University of Manchester, University of California, Irvine
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223111950