A connection between low-lying zeros and central values of $L$-functions

TL;DR

This paper explores the link between low-lying zeros of $L$-functions and their central values, deriving bounds related to prominent conjectures by connecting symmetry types and Fourier support.

Contribution

It establishes a connection between the statistics of low-lying zeros and the distribution of central $L$-values, providing explicit bounds and relating symmetry types to the quality of these bounds.

Findings

Derived explicit conditional lower bounds for the Keating-Snaith conjecture.

Identified the common ingredient in approaches to the Rudnick-Sarnak and Keating-Snaith conjectures.

Connected symmetry types, Fourier support, and bounds quality in $L$-function families.

Abstract

We discuss the relation between statistics on low-lying zeros of $L$-functions and distribution of the associated central values. More precisely, we deduce explicit conditional lower bounds toward the Keating-Snaith conjecture (on the distribution of central values of families of $L$-functions) from partial results toward the Rudnick-Sarnak density conjecture (on the one-level density for the low-lying zeros of these $L$-functions). We show in fact that the same crucial ingredient occurs in the classical approaches for proving both results, providing the connection. We precisely determine the relation between the type of symmetry of the family, the allowed Fourier support in its distributional statement, and the quality of the lower bounds obtained.