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

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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 $L\left(s,\pi \right)$ be an automorphic L-function of $GL\left(n\right)$, where $\pi$ is an automorphic representation of group $GL\left(n\right)$ over rational number field $\mathbb{Q}$. In this paper, we study the zero-density estimates for $L\left(s,\pi \right)$. Define ${N}_{\pi }\left(\sigma ,{T}_{1},{T}_{2}\right)$ = ♯ {$\rho$ = $\beta$ + $i\gamma$: $L\left(\rho ,\pi \right)$ = 0, $\sigma <\beta <1$, ${T}_{1}\le \gamma \le {T}_{2}$}, where $0\le \sigma <1$ and ${T}_{1}<{T}_{2}$. We first establish an upper bound for ${N}_{\pi }\left(\sigma ,T,2T\right)$ when $\sigma$ is close to 1. Then we restrict the imaginary part $\gamma$ into a narrow strip $\left[T,T+{T}^{\alpha }\right]$ with $0<\alpha \le 1$ and prove some new zero-density results on ${N}_{\pi }\left(\sigma ,T,T+{T}^{\alpha }\right)$ under specific conditions, which improves previous results when $\sigma$ near $\frac{3}{4}$ and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method. View Full-Text
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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