New Zero-Density Results for Automorphic <i>L</i>-Functions of <i>GL</i>(<i>n</i>)

Abstract

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, σβ1, T1≤γ≤T2}, where 0≤σ1 and T1T2. 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 0α≤1 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.

Keywords

• zero density
• automorphic L-function
• automorphic representation