Extending the unconditional support in an Iwaniec-Luo-Sarnak family

TL;DR

This paper extends the support range for the one-level density of low-lying zeros in a family of holomorphic newforms, approaching the GRH limit by employing zero-density estimates for Dirichlet L-functions.

Contribution

It advances the understanding of low-lying zeros by enlarging the support range in the harmonic analysis of L-functions, nearing the GRH boundary.

Findings

Support extended to (-Θ_k, Θ_k) with Θ_k approaching 2 as k increases

Achieved support range asymptotically as good as the best GRH results

Utilized zero-density estimates for Dirichlet L-functions in the analysis

Abstract

We study the harmonically weighted one-level density of low-lying zeros of $L$-functions in the family of holomorpic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz--Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this admissible support to $(-\Theta_k,\Theta_k)$, where $\Theta_2 = 1.866\dots$ and $\Theta_k$ tends monotonically to $2$ as $k$ tends to infinity. This is asymptotically as good as the best known GRH result. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions.