Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
Abstract
For T > 0 large, let τ be uniformly distributed on [T,2T] and let σ= σ(T) be a deterministic real part approaching 1/2 from the right. For the Riemann zeta function ζ, we study the Gibbs measure with density proportional to |ζ(σ+ iτ+ i h)|β on Iθ := [ − (logT)θ,(logT)θ], for every θ> 0 and β> 0. As β increases, the Gibbs measure increasingly favors positions where |ζ| is large, providing a way to study the spatial organization of extreme values. Under explicit conditions on the rate at which σ approaches the critical line, a phase transition is established at the critical inverse temperature βc(θ) := 2√(1 + θ). The limiting distribution of the correlation (or overlap) between the field values at two positions sampled from the Gibbs measure is concentrated at $0$ when β≤ βc(θ), and assigns masses βc(θ)/β and 1 − βc(θ)/β to $0$ and $1$, respectively, when β> βc(θ). Joint convergence of the overlaps is also established, with the limiting arrays described by Poisson-Dirichlet weights in the low-temperature regime. The Ghirlanda-Guerra identities, which provide recursive relations for the joint overlap distributions, are established in the limit, uniformly for bounded deterministic observables of the sampled positions. All our results are established independently of the Riemann hypothesis.
Related
- Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line
- Simple critical zeros and distinct zeros of the Riemann zeta function in short intervals
- Large value estimates for Dirichlet polynomials and trilinear sums over short intervals
- Convex Bernstein limits and sharp recovery on expanding intervals
- Almost primes in short intervals and averages of long Dirichlet polynomial approximations to (ζ'/ζ)k
BibTeX
@misc{ouimet2026poissondirichlet,
title = {{Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line}},
author = {Frédéric Ouimet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22821},
eprint = {2610.22821},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22821},
}
RIS
TY - UNPB
TI - Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line
AU - Frédéric Ouimet
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22821
UR - https://arxiv.org/abs/2610.22821
AB - For T > 0 large, let τ be uniformly distributed on [T,2T] and let σ= σ(T) be a deterministic real part approaching 1/2 from the right. For the Riemann zeta function ζ, we study the Gibbs measure with density proportional to |ζ(σ+ iτ+ i h)|^β on I_θ := [ − (logT)^θ,(logT)^θ], for every θ> 0 and β> 0. As β increases, the Gibbs measure increasingly favors positions where |ζ| is large, providing a way to study the spatial organization of extreme values. Under explicit conditions on the rate at which σ approaches the critical line, a phase transition is established at the critical inverse temperature β_c(θ) := 2√(1 + θ). The limiting distribution of the correlation (or overlap) between the field values at two positions sampled from the Gibbs measure is concentrated at $0$ when β≤ β_c(θ), and assigns masses β_c(θ)/β and 1 − β_c(θ)/β to $0$ and $1$, respectively, when β> β_c(θ). Joint convergence of the overlaps is also established, with the limiting arrays described by Poisson-Dirichlet weights in the low-temperature regime. The Ghirlanda-Guerra identities, which provide recursive relations for the joint overlap distributions, are established in the limit, uniformly for bounded deterministic observables of the sampled positions. All our results are established independently of the Riemann hypothesis.
ER -