Hilbert spaces and low-lying zeros of L-functions

TL;DR

This paper develops a unified framework using reproducing kernel Hilbert spaces to analyze low-lying zeros of L-functions, providing explicit formulas and solving related Fourier optimization problems across different symmetry types.

Contribution

It introduces a novel approach based on reproducing kernel Hilbert spaces to estimate zeros of L-functions and solves key optimization problems explicitly.

Findings

Explicit reproducing kernels for symmetry types

Unified framework for low-lying zero analysis

Connections to de Branges spaces and Paley-Wiener space

Abstract

Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line (measured by the analytic conductor). To solve the Fourier optimization problems that arise, we provide a unified framework based on the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type). Explicit expressions for the reproducing kernels are given. We also revisit the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). We solve the associated Fourier optimization problem in this setting by establishing a connection to the theory of de Branges spaces of entire functions and using the explicit reproducing kernels. In an appendix, we study the related problem of determining the sharp embeddings between the Hilbert spaces associated to the five symmetry types and the classical Paley-Wiener space.