The Omega Result Of Coefficients Of Automorphic L-functions

In generally, in term with Langlands Programs, there are many hidden structures underlying the Fourier coefficients of an automor-phic form. Any most general L-function should be a product of L-functions of automorphic cuspidal representations of GLm/Q. Other parts of the Langlands conjectures imply that the Ramanujan-Petersson conjectures should hold for any automorphic L-function. Thus it is very important and essential to investigate the automorphic L-functions. We introduce some basic facts about the holomorphic cusp forms and their L-functions. Proofs for these results can be found. The full modular group Γ=SL2(Z)Let f∈Sk(Γ) is an eigenfunction of all the Hecke operators we shall write Tn formally aswhere Hecke operators bewhere λf(n) have properties:(3) for all positive integers m>1,n>1,we haveLet Hk*be the set of all normalized Hecke eigencuspforms of weight k for Γ=SL2(Z).The Hecke L-function attached to f∈Hk*is defined, The size and oscillations of λ∫(n)are interesting objects.In paper,we study the omega result of the error term of the asymptotic formula of and we establish the following some theorems.Theorem1Let f∈Hk*,and λf(n)denote its n-th normalized Fourier coefficient.where c is a suitable constant.then,we haveTheorem2Let f∈Hk*,and λf(n)denote its n-th normalized Fourier coefficient,where c1is a suitable constant.then,we haveTheorem3Let f∈Hk*,and λf(n)denote its n-th normalized Fourier coefficient,where c2is a suitable constant.then,we have...