Zero Density Estimates Of Automorphic L-functions Associated With Maass Forms Twist By Character

In general, An L-function is a generating function defined by a Dirichlet series. According to Langlands Program, any " most general" L-function(formed out of local data such as arithmetic or geometric) shoud be a product of L-funtions of automorphic cuspidal representations of GL(m). Among the theory of automorphic L-functions, The Generalized Riemann Hypothesis (GRH) resides in the core status, namely all the non-trivial zeros of automorphic L-function lie on the critical line Res=1/2. In the absence of GRH, it is natural to consider how many zeros of L-function lie off the critical line. This is the problem of zero density estimates for L-function.The problem of zero density estimates for L-function is one of the most important problems in,analytic number theory, which has wide applications such as the prime dis-tribution in short interval and the least prime in arithmetic progressions. Thus it is very important and issential to investigate the problem of zero density estimates for L-function. Some people studied zero density estimates of automorphic L-functions associated with holomorphic cusp forms.Define where 1/2≤σ≤1, T≥3,ρare the zeros of automorphic L-function L(s,f (?)χ)= and a(n) is the Fourier coefficient of holomorphic cusp forms. n Ivic [1]obtained that Kamiya [2] and Dong Xinmei [3] proved thatZhang Deyu [4] obtained thatwhereLet f be a normalized Maass cusp form for the full modular group SL2(Z), and f is the eigenfunction of Hecke operator T(n):T(n)f=λ(n)f. Then we have A(1)=1. DefinewhereA.Sankaranaranayanan and J.Sengupta [5] proved thatXu Zhao [6]proved thatIn this paper,we will investigate the zero density estimates of automorphic L-functions associated with Maass form. Define: We want to prove that the zero density estimates as follows: If q=1, the above formula is the problem of zero density for automorphic L-function L(s,f). It is generally conjectured that the above formula holds for A(σ)=2,1/2≤σ≤1. Ingham [7] and Huxley [8] proved that if A(σ)=12/5, (0.3)is correct.In this paper,we will obtain the following results:Theorem 1. For N(σ,T,χ) defined by (0.1), we haveTheorem 2. For N(σ,T,χ) defined by (0.1), we have...