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

TL;DR

This paper proves Montgomery's pair correlation conjecture for the zeros of Hardy’s Z-function assuming the Riemann Hypothesis, by modeling zeros as a real hall akin to a random matrix ensemble and demonstrating their universal GUE statistics.

Contribution

It introduces a finite-dimensional variational space and a stochastic dynamics approach to establish the GUE statistics of Hardy Z-function zeros under the Riemann Hypothesis.

Findings

Zeros of Hardy's Z-function exhibit GUE sine-kernel statistics.

The dynamics of zeros follow Dyson Brownian motion with =2.

Pair-correlation measures converge to the GUE law.

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 $\beta=2$, 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 $Z(t)$.