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

Abstract

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,β)$ defined to be the number of pairs $γ$ and $γ'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<γ,γ'\leq T$ and $0 < γ'-γ\leq 2πβ/\log T$ as $T\to \infty$. In this paper, assuming the Riemann hypothesis, we prove upper and lower bounds for $N(T,β)$, for all $β>0$, 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 $[-β, β]$ in a way to minimize the $L^1\big(\mathbb{R}, \big\{1 - \big(\frac{\sin πx}{πx}\big)^2 \big\}\,dx\big)$-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 $β\in \frac12 \mathbb{N}$ was considered using non-extremal majorants and minorants.