Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line

Abstract

Let T → ∞, let τ be uniformly distributed on [T,2T], and let σ= σ(T) be a deterministic real part approaching 1/2 from the right. We establish the convergence in law of the recentered maximum of log|ζ(σ+ iτ+ i h)| over I_θ = smash[ − (log T)^θ,(log T)^θ], for every fixed θ> 0. Under explicit conditions on the displacement $σ- 1/2$, the recentered maximum converges to a Gumbel distribution with scale parameter 1/(2√(1 + θ)). The recentering is an exact upper quantile of the one-point law of the associated Haar field F_h^H = ℜ∑_{p^j≤ X}U_p^j/(jp^{j(σ+ i h)}), where the U_p are independent and uniformly distributed on the complex unit circle. The positive constant in the limiting distribution is characterized by a Gaussian process describing the limiting microscopic increments of the Haar field around a high point. The result is established without the Riemann hypothesis.

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BibTeX RIS

BibTeX

@misc{ouimet2026convergence,
  title = {{Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line}},
  author = {Frédéric Ouimet},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01905},
  eprint = {2611.01905},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01905},
}

RIS

TY  - UNPB
TI  - Convergence in law of the maximum of the Riemann zeta function on expanding intervals near the critical line
AU  - Frédéric Ouimet
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01905
UR  - https://arxiv.org/abs/2611.01905
AB  - Let T → ∞, let τ be uniformly distributed on [T,2T], and let σ= σ(T) be a deterministic real part approaching 1/2 from the right. We establish the convergence in law of the recentered maximum of log|ζ(σ+ iτ+ i h)| over I_θ = smash[ − (log T)^θ,(log T)^θ], for every fixed θ> 0. Under explicit conditions on the displacement $σ- 1/2$, the recentered maximum converges to a Gumbel distribution with scale parameter 1/(2√(1 + θ)). The recentering is an exact upper quantile of the one-point law of the associated Haar field F_h^H = ℜ∑_p^j≤ XU_p^j/(jp^j(σ+ i h)), where the U_p are independent and uniformly distributed on the complex unit circle. The positive constant in the limiting distribution is characterized by a Gaussian process describing the limiting microscopic increments of the Haar field around a high point. The result is established without the Riemann hypothesis.
ER  -

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