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Brownian behaviour of the Riemann zeta function around the critical line

Louis Vassaux

TL;DR

This work proves a functional Brownian limit for the logarithm of the Riemann zeta function along horizontal perturbations near the critical line: the process $Z^{(T)}(\alpha) = (1/\sqrt{\log \log T}) \log \zeta(1/2 + 1/(\log T)^{\alpha} + i\tau)$ on $[0,1]$ converges in law to a standard complex Brownian motion as $T\to\infty$. The proof combines finite-dimensional Gaussian convergence (via Dirichlet polynomial approximations and Bourgade-type lemmas) with a Kolmogorov-type tightness argument, where zeros of $\zeta$ are carefully controlled. Consequences include a Brownian reflection principle for the maximum of $\log|\zeta|$ on horizontal lines, an arcsine law for the horizontal distribution of large values, and a Brownian-type law of the iterated logarithm for horizontal drift, together with local-time and occupation-measure descriptions of $\log|\zeta|$. These results deepen the probabilistic understanding of zeta-values in the critical strip and reveal a Brownian structure underlying horizontal fluctuations, paralleling random matrix and stochastic-process models.

Abstract

We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for $ζ$, including an analogue of the reflection principle for the maximum of the Brownian motion: as $T$ diverges, for any $u>0$ we have \[ \frac{1}{T}\cdot {\rm meas}\Big\{0\leq t\leq T:\max_{σ\geq \tfrac{1}{2}}\log|ζ(σ+i t)|\geq u \sqrt{\tfrac{1}{2}\log \log T} \Big\}\to 2 \displaystyle\int_u^{\infty} \frac{e^{-\frac{x^2}{2}}}{\sqrt{2π}}\mathrm{d} x. \]

Brownian behaviour of the Riemann zeta function around the critical line

TL;DR

This work proves a functional Brownian limit for the logarithm of the Riemann zeta function along horizontal perturbations near the critical line: the process on converges in law to a standard complex Brownian motion as . The proof combines finite-dimensional Gaussian convergence (via Dirichlet polynomial approximations and Bourgade-type lemmas) with a Kolmogorov-type tightness argument, where zeros of are carefully controlled. Consequences include a Brownian reflection principle for the maximum of on horizontal lines, an arcsine law for the horizontal distribution of large values, and a Brownian-type law of the iterated logarithm for horizontal drift, together with local-time and occupation-measure descriptions of . These results deepen the probabilistic understanding of zeta-values in the critical strip and reveal a Brownian structure underlying horizontal fluctuations, paralleling random matrix and stochastic-process models.

Abstract

We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for , including an analogue of the reflection principle for the maximum of the Brownian motion: as diverges, for any we have
Paper Structure (13 sections, 17 theorems, 85 equations)

This paper contains 13 sections, 17 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Statistics on $\zeta$.
  3. Main result.
  4. Consequences.
  5. Notations and acknowledgments
  6. Overview of the proof of Theorem \ref{['mainthm']}
  7. Convergence of finite-dimensional distributions.
  8. The tightness criterion.
  9. Structure of the proof of tightness
  10. Moments of Dirichlet sums
  11. The contribution of zeta zeroes
  12. Bounding the error $e_x$.
  13. Towards the critical line

Key Result

Theorem 1.1

Let $T > 10$ an $\tau$ be a uniform random variable on $[0, T]$. Define Then, as $T \rightarrow \infty$, $Z^{(T)}$ converges in law, for the topology of uniform convergence in $\mathcal{C}^0([0,1], \mathbb{C})$, to a standard complex Brownian motion $B$, i.e. $B=\tfrac{1}{\sqrt{2}}(B_1+iB_2)$ where $B_1$ and $B_2$ are independent, standard real Brownian motions.

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3: Reflection principle
  • proof
  • Corollary 1.4: Arcsine law
  • proof
  • Corollary 1.5: Law of the iterated logarithm
  • proof
  • Corollary 1.6
  • proof
  • ...and 19 more