Zero Density Estimates For Automorphic L-Functions

An L-function is a type of generating function formed out of local data associated with either an arithmetic-geometric object ( such as an abclian variety defined over a number field ) or with an automorphic form. According to conjectures in the Langlands Program, any "most general" L-function should be a product of L-functions of automorphic cuspidal representations of GL_m/Q. Other parts of the Langlands conjectures imply that the Ramanujan-Petersson conjecture should hold for any automorphic L-function. Thus it is very important and essential to investigate the analytic properties of these automorphic L-functions.In this thesis, we study automorphic L-functions attached to holomorphic cusp forms of SL₂/Z.Let f be a holomorphic cusp form for the group Γ = SL₂(Z) of even integral weight k, with Fourier coefficients α_f(n):We normalize f by setting α_f(1) = 1 and set λ_f(n) = α_f(n)/n^(k-1)/2. It follows from the Ramanujan Conjecture thatwhere d{n) is the divisor function. See e.g. [10], (14.54).Letwhere λ(modq) is a primitive character and q varies over the segment 1 ≤ q ≤ Q.