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New Zero-Density Results for Automorphic L-Functions of GL(n)

School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, China
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Author to whom correspondence should be addressed.
Academic Editors: Diana Savin and Alexander Felshtyn
Mathematics 2021, 9(17), 2061; https://doi.org/10.3390/math9172061
Received: 17 June 2021 / Revised: 11 August 2021 / Accepted: 24 August 2021 / Published: 26 August 2021
Let L(s,π) be an automorphic L-function of GL(n), where π is an automorphic representation of group GL(n) over rational number field Q. In this paper, we study the zero-density estimates for L(s,π). Define Nπ(σ,T1,T2) = ♯ {ρ = β + iγ: L(ρ,π) = 0, σ<β<1, T1γT2}, where 0σ<1 and T1<T2. We first establish an upper bound for Nπ(σ,T,2T) when σ is close to 1. Then we restrict the imaginary part γ into a narrow strip [T,T+Tα] with 0<α1 and prove some new zero-density results on Nπ(σ,T,T+Tα) under specific conditions, which improves previous results when σ near 34 and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method. View Full-Text
Keywords: zero density; automorphic L-function; automorphic representation zero density; automorphic L-function; automorphic representation
MDPI and ACS Style

Ding, W.; Liu, H.; Zhang, D. New Zero-Density Results for Automorphic L-Functions of GL(n). Mathematics 2021, 9, 2061. https://doi.org/10.3390/math9172061

AMA Style

Ding W, Liu H, Zhang D. New Zero-Density Results for Automorphic L-Functions of GL(n). Mathematics. 2021; 9(17):2061. https://doi.org/10.3390/math9172061

Chicago/Turabian Style

Ding, Wenjing, Huafeng Liu, and Deyu Zhang. 2021. "New Zero-Density Results for Automorphic L-Functions of GL(n)" Mathematics 9, no. 17: 2061. https://doi.org/10.3390/math9172061

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