Conditional lower bounds on the distribution of central values in families of $L$-functions
Volume 214 / 2024
Acta Arithmetica 214 (2024), 481-497
MSC: Primary 11M41; Secondary 11G05
DOI: 10.4064/aa230805-3-1
Published online: 7 March 2024
Abstract
We establish a general principle that any lower bound on the non-vanishing of central $L$-values obtained through studying the one-level density of low-lying zeros can be refined to show that most such $L$-values have the typical size conjectured by Keating and Snaith. We illustrate this technique in the case of quadratic twists of a given elliptic curve, and similar results should hold for the many examples studied by Iwaniec, Luo, and Sarnak in their pioneering work (2000) on $1$-level densities.
