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 $N(T,\beta )$ defined to be the number of pairs $\gamma$ and ${\gamma }^{\prime }$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying $0<\gamma ,{\gamma }^{\prime }\le T$ and $0<{\gamma }^{\prime }-\gamma \le 2\pi \beta /\log T$ 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 $L^1(\mathbb{R},\{1-\left(\frac{\sin \pi x}{\pi x}{)}^2\right\}dx)$-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 $\beta \in \frac{1}{2}\mathbb{N}$ was considered using non-extremal majorants and minorants.

to appear in J. Reine Angew. Math

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• Bounding $\zeta (s)$ in the critical strip
• Extremal functions in de Branges and Euclidean spaces
• Asymptotic Large Sieve
• Extremal problems in de Branges spaces: the case of truncated and odd functions

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