Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function

TL;DR

This paper establishes bounds on the pair correlation of zeros of the Riemann zeta-function assuming the Riemann hypothesis, using extremal functions and reproducing kernel Hilbert spaces to extend previous results.

Contribution

It provides a complete solution to the extremal problem for bounding pair correlations, extending Gallagher's earlier work to all positive eta, using optimal majorants and minorants.

Findings

Derived upper and lower bounds for $N(T,eta)$ for all eta > 0.

Extended previous work by Gallagher to all eta, not just half-integers.

Utilized extremal functions of exponential type within reproducing kernel Hilbert spaces.

Abstract

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<\gamma,\gamma'\leq T$ and $0 < \gamma'-\gamma \leq 2\pi \beta/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,\beta)$, for all $\beta >0$, using Montgomery's formula and some extremal functions of exponential type. These functions are optimal in the sense that they majorize and minorize the characteristic function of the interval $[-\beta, \beta]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin \pi x}{\pi x}\big)^2 \big\}\,dx\big)$-error. We give a complete solution for this extremal problem using the framework of reproducing kernel Hilbert spaces of entire functions. This extends previous work by P. X. Gallagher in 1985, where the case $\beta \in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.