Some Problems On Coefficients Of L-functions

L-functions appear to be an esoteric and special topic in number theory.The most classical one is Riemann zeta function.Similar to Riemann zeta function,there exist grand Riemann hypothesis,grand Ramanujan conjecture and so on for the general L-functions.It is well known that grand Ramanujan conjecture is still open in most cases.In this thesis,without assuming the generalized Ramanujan conjecture,we study some distribution laws for coefficients of L-functions.In chapter 1,we establish a general summation formula for the coefficients of a class of L-functions,without assuming the generalized Ramanujan conjecture.As an application,we consider integral power moments of Fourier coefficients of Hecke-Maass cusp forms.In chapter 2,We focus our attention on automorphic L-functions.Let ? be an unitary cuspidal automorphic representation for GLm(AQ),and let L(s,?)be the automorphic L-function attached to ?,which has a Dirichlet series expression in the half-plane Rs>1,i.e.We are interested in the upper bound of the fourth power moment of ??(n),i.e.?n?x|??(n)|4.If m= 2,we are able to consider the sixteenth power moment of??(n).As an application,we consider the lower bound of ?n?x|??(n)|,which im-proves previous results.In chapter 3,we study the analogue of the Bombieri-Vinogradov theorem for SLm(Z)Hecke-Maass form F(z).In particular,for SL2(Z)holomorphic or Maass Hecke eigenforms,symmetric-square lifts of holomorphic Hecke eigenforms on SL2(Z),and SL3(Z)Maass Hecke eigenforms under the Ramanujan conjecture,the levels of distribution are all equal to 1/2,which is as strong as the Bombieri-Vinogradov theorem.As an application,we give some savings for shifted convolu-tion sums at primes,namely for a ? 0,(?)(?)where p(n)are Fourier coefficients ?f(n)of a holomorphic Hecke eigenform f for SL2(Z)or Fourier coefficients AF(n,1)of its symmetric-square lift F.Further,as a consequence,we get an asymptotic formula(?)(?)where E1(a)is a constant depending on a.