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

Let L ( s , π ) be an automorphic L -function of G L ( n ) , where π is an automorphic representation of group G L ( n ) over rational number field Q . In this paper, we study the zero-density estimates for L ( s , π ) . Define N π ( σ , T 1 , T 2 ) = ♯ { ρ = β + i γ : L ( ρ , π ) = 0, σ < β < 1 , T 1 ≤ γ ≤ T 2 }, where 0 ≤ σ < 1 and T 1 < T 2 . We first establish an upper bound for N π ( σ , T , 2 T ) when σ is close to 1. Then we restrict the imaginary part γ into a narrow strip [ T , T + T α ] with 0 < α ≤ 1 and prove some new zero-density results on N π ( σ , T , T + T α ) under specific conditions, which improves previous results when σ near 3 4 and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method.