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Variations of the Hardy Z-Function and the Montgomery Pair Correlation Conjecture

Yochay Jerby

TL;DR

The paper delivers an intrinsic analytic mechanism linking the Riemann zeta zeros to random-matrix statistics by constructing a finite-dimensional analytic ensemble RH_N(R) of Hardy Z-function approximants and evolving their zeros via a Skorokhod SDE with boundary reflection. Through Dyson Brownian motion universality for local statistics and Selberg-type decorrelation across windows, it first proves an ensemble GUE pair-correlation for the zeros of Z_N(t; a) and then transfers this to a deterministic, macroscopic setting for Hardy’s Z-function, establishing Montgomery’s pair-correlation conjecture under RH. The key strategy combines an internal analytic ensemble with stochastic dynamics, whitening techniques, and Girsanov-type measure changes to show that the microscopic zero interactions replicate the sine-kernel of GUE, while macroscopic decorrelation yields ergodicity of pair-correlations. This work thus provides a rigorous, internally motivated mechanism for GUE statistics in zeta zeros, bridging deterministic zeta structure with universal random-matrix behavior and offering a pathway to understanding quantum-chaos-type statistics from analytic number theory.

Abstract

In 1973 Montgomery formulated the pair correlation conjecture, predicting that the local spacing statistics of the nontrivial zeros of the Riemann zeta function coincide with those of eigenvalues of large Hermitian matrices from the Gaussian Unitary Ensemble (GUE). The zeta function, however, is a fixed deterministic object, and the mechanism by which its zeros reproduce random matrix statistics has remained unclear. In this paper, assuming the Riemann Hypothesis, we prove Montgomery's pair correlation conjecture for the zeros of Hardy's $Z$-function. Building on earlier works, we use a finite-dimensional variational space of sections $Z_N(t;a)$ that approximate $Z(t)$ on each window $[2N,2N+2]$. Inside this space we define the real hall $\mathcal{RH}_N(\mathbb{R})$, consisting of those sections whose zeros in the corresponding critical rectangle are real, simple, and remain so along any homotopy from the core section. This real hall plays the role of a random matrix ensemble. Equipping it with an admissible probability measure with smooth, positive density on the coefficient space, we construct a Skorokhod-type stochastic differential equation with reflection at the discriminant boundary. We show that the induced dynamics of the unfolded zeros are, in the bulk, equivalent in law to Dyson Brownian motion with $β=2$, and by invoking modern universality results for Dyson Brownian motion and log-gases we obtain that, for any such measure, the ensemble-averaged local pair-correlation converges to the GUE sine-kernel law. Finally, using Selberg's probabilistic theory of the argument $S(t)$, we prove that the pair-correlation observables of the canonical approximants $Z_N(t;1)$ over disjoint windows behave asymptotically like decorrelated samples drawn from this GUE-distributed ensemble, and we upgrade the averaged GUE law to a deterministic pair-correlation law for the zeros of $Z(t)$.

Variations of the Hardy Z-Function and the Montgomery Pair Correlation Conjecture

TL;DR

The paper delivers an intrinsic analytic mechanism linking the Riemann zeta zeros to random-matrix statistics by constructing a finite-dimensional analytic ensemble RH_N(R) of Hardy Z-function approximants and evolving their zeros via a Skorokhod SDE with boundary reflection. Through Dyson Brownian motion universality for local statistics and Selberg-type decorrelation across windows, it first proves an ensemble GUE pair-correlation for the zeros of Z_N(t; a) and then transfers this to a deterministic, macroscopic setting for Hardy’s Z-function, establishing Montgomery’s pair-correlation conjecture under RH. The key strategy combines an internal analytic ensemble with stochastic dynamics, whitening techniques, and Girsanov-type measure changes to show that the microscopic zero interactions replicate the sine-kernel of GUE, while macroscopic decorrelation yields ergodicity of pair-correlations. This work thus provides a rigorous, internally motivated mechanism for GUE statistics in zeta zeros, bridging deterministic zeta structure with universal random-matrix behavior and offering a pathway to understanding quantum-chaos-type statistics from analytic number theory.

Abstract

In 1973 Montgomery formulated the pair correlation conjecture, predicting that the local spacing statistics of the nontrivial zeros of the Riemann zeta function coincide with those of eigenvalues of large Hermitian matrices from the Gaussian Unitary Ensemble (GUE). The zeta function, however, is a fixed deterministic object, and the mechanism by which its zeros reproduce random matrix statistics has remained unclear. In this paper, assuming the Riemann Hypothesis, we prove Montgomery's pair correlation conjecture for the zeros of Hardy's -function. Building on earlier works, we use a finite-dimensional variational space of sections that approximate on each window . Inside this space we define the real hall , consisting of those sections whose zeros in the corresponding critical rectangle are real, simple, and remain so along any homotopy from the core section. This real hall plays the role of a random matrix ensemble. Equipping it with an admissible probability measure with smooth, positive density on the coefficient space, we construct a Skorokhod-type stochastic differential equation with reflection at the discriminant boundary. We show that the induced dynamics of the unfolded zeros are, in the bulk, equivalent in law to Dyson Brownian motion with , and by invoking modern universality results for Dyson Brownian motion and log-gases we obtain that, for any such measure, the ensemble-averaged local pair-correlation converges to the GUE sine-kernel law. Finally, using Selberg's probabilistic theory of the argument , we prove that the pair-correlation observables of the canonical approximants over disjoint windows behave asymptotically like decorrelated samples drawn from this GUE-distributed ensemble, and we upgrade the averaged GUE law to a deterministic pair-correlation law for the zeros of .

Paper Structure

This paper contains 29 sections, 37 theorems, 454 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Montgomery Pair Correlation Conjecture and Random Matrix Theory
  3. The Microscopic Question - Dyson Brownian Motion on Zeros of $\mathcal{RH}_N(\mathbb{R})$
  4. The Macroscopic Question - the Ergodicity of the Pair-Correlations of $Z_N(t; \bar{1})$
  5. Organization of the Paper
  6. Review of GUE Eigenvalue Statistics and Universality for Dyson's 1D Coulomb Gas Model
  7. Random Matrices and 1D Coulomb Gas
  8. Itô’s Calculus and Dyson Brownian Motion for Hermitian Matrices
  9. Modern universality theory for DBM
  10. The Skorokhod SDE on $\mathcal{RH}_N (\mathbb{R})$ and the Induced SDEs on zeros
  11. The Quadratic Co-Variation Matrix of the First-Order Noises
  12. Normalized First-Order Noises are Asymptotically Independent Brownian Motions
  13. Zeros of Hadamard Product and Coulomb-Type Repulsion
  14. Universality and Reduction of the SDE to the Classical DBM
  15. Definition of the drifts $b^{reg}_n,b^{err}_n, b_n^{Sko}$ and $b^{1-body}_n$
  16. ...and 14 more sections

Key Result

Theorem A

As $N \to \infty$, the average pair correlation of the zeros of sections $Z_N(t;\bar{a}) \in \mathcal{RH}_N(\mathbb{R})$, taken with respect to any admissible probability measure $\mu_N$ on $\mathcal{RH}_N(\mathbb{R})$ with a smooth, strictly positive density with respect to the Lebesgue measure on where $M_N \sim \tfrac{1}{\pi}\log\!\bigl(\tfrac{N}{\pi}\bigr)$ is the number of zeros $\{t_j(\bar{

Figures (1)

  • Figure 1: Histogram of nearest-neighbour spacings of the $164$ zeros of $Z_{N}(t;\bar{a})$, with $N=300$, in $755\le t\le 965$ where $\bar{a} \in \mathbb{R}^N$ is randomized so that its zeros in this region are real, and the GUE Wigner surmise $P_{\text{GUE}}(s)=\tfrac{32}{\pi^{2}}s^{2}e^{-4s^{2}/\pi}$ (red).

Theorems & Definitions (91)

  • Definition
  • Theorem A: GUE law for the ensemble $\mathcal{RH}_N(\mathbb{R})$
  • Theorem B: GUE Pair--Correlation for randomized $Z$--functions
  • Theorem C: Montgomery Pair--Correlation Theorem for Hardy’s $Z$--function
  • Remark 2.1: Collisions occur with probability zero
  • Definition 3.1: The $N$-th real hall
  • Remark 3.1: $\mathcal{RH}_N(\mathbb{R})$ and the RH
  • Remark 3.2: The geometry of the boundary $\partial \mathcal{RH}_N (\mathbb{R})$
  • Proposition 3.1
  • Remark 3.3: RMT analogy of $\mathcal{RH}_N (\mathbb{R}) \subset \mathcal{Z}_N(\mathbb{R})$
  • ...and 81 more