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

Journal article, 2025

We study the harmonically weighted one-level density of low-lying zeros of L-functions in the family of holomorphic 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 (formula presenetd). This result was improved by Ricotta–Royer, who increased the admissible support for k ≥ 4 in a way that is asymptotically as good as the best known GRH result. We extend the admissible support for all (formula presenetd), where ‚(formula presenetd) 1:866 ... and (formula presenetd) tends monotonically to 2 asymptotically five times faster than what was previously known. The main novelty in our analysis is the use of zero-density estimates for Dirichlet L-functi