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

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.Comment: to appear in J. Reine Angew. Mat

The adaptive Levin method

The Levin method is a classical technique for evaluating oscillatory integrals that operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that the method suffers from ``low-frequency breakdown,'' meaning that the accuracy of the computed integral deteriorates when the integrand is only slowly oscillating. Recently presented experimental evidence suggests that, when a Chebyshev spectral method is used to discretize the differential equation and the resulting linear system is solved via a truncated singular value decomposition, no such phenomenon is observed. Here, we provide a proof that this is, in fact, the case, and, remarkably, our proof applies even in the presence of saddle points. We also observe that the absence of low-frequency breakdown makes the Levin method suitable for use as the basis of an adaptive integration method. We describe extensive numerical experiments demonstrating that the resulting adaptive Levin method can efficiently and accurately evaluate a large class of oscillatory integrals, including many with saddle points