PreprintCC BY 4.0arXiv
Zero density estimates for GL2 automorphic L-functions
Abstract
We prove zero density estimates for L-functions of cuspidal automorphic representations π of GL2(𝔸ℚ). We show that Nπ(σ, T) ≪ T5/2(1 − σ) + o(1), where Nπ(σ, T) denotes the number of zeros ρ= β+ iγ of L(s,π) with β≥ σ and |γ| ≤ T. The key input is an extension of the Guthunicodex2013Maynard argument to Dirichlet polynomials whose coefficients satisfy a weaker ℓq bound.
Related
- A zero-density estimate for the symmetric square L-functions in the weight aspect
- Surfaces Associated to Zeros of Automorphic L-functions
- Lower Bounds for Localised Moments of Automorphic L-functions
- Probing automorphic induction through zero correlations and secondary arithmetic terms
- Schauder estimates for ratios of elliptic solutions under zero set inclusion
BibTeX
@misc{cossaboom2026zero,
title = {{Zero density estimates for $\mathrmGL_2$ automorphic $L$-functions}},
author = {Catherine Cossaboom},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27715},
eprint = {2609.27715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27715},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Zero density estimates for GL_2 automorphic L-functions
AU - Catherine Cossaboom
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27715
UR - https://arxiv.org/abs/2609.27715
AB - We prove zero density estimates for L-functions of cuspidal automorphic representations π of GL_2(𝔸_ℚ). We show that N_π(σ, T) ≪ T^5/2(1 − σ) + o(1), where N_π(σ, T) denotes the number of zeros ρ= β+ iγ of L(s,π) with β≥ σ and |γ| ≤ T. The key input is an extension of the Guthunicodex2013Maynard argument to Dirichlet polynomials whose coefficients satisfy a weaker ℓ^q bound.
ER -