Upper bounds for the lowest first zero in families of cuspidal newforms

Authors

Palak Arora, Williams College
Glenn Bruda, University of Florida
Bruce Fang, Williams College
Raul Marquez, The University of Texas Rio Grande Valley
Steven J. Miller, Williams College
Beni Prapashtica, University of Cambridge
Vismay Sharan, Yale University
Daeyoung Son, Williams College
Xueyiming Tang, The Loomis Chaffee School
Saad Waheed, Williams College

Document Type

Article

Publication Date

12-2025

Abstract

Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of L-functions lie on the critical line with the real part 1/2. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the n-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within 0.218503 of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point.

Comments

Student publication. Original published version available at https://doi.org/10.1016/j.jnt.2025.02.012

Recommended Citation

Arora, Palak, Glenn Bruda, Bruce Fang, et al. “Upper Bounds for the Lowest First Zero in Families of Cuspidal Newforms.” Journal of Number Theory 277 (December 2025): 262–89. https://doi.org/10.1016/j.jnt.2025.02.012.

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Journal of Number Theory

DOI

10.1016/j.jnt.2025.02.012