Integral Mean Value Of Automorphic L-function On GL₃

The twentieth century is a period of rapid development of number theory,especially the theory of automorphic form and automorphic representation.Langlands program connects the arithmetic problem of algebraic variety with automorphic representation.Most of the advanced topics in modern number theory revolve around the Langlands program.Automorphic formal theory is the core content of Langlands program,which is a new field at the intersection of number theory,algebra,harmonic analysis and geometry.It puts forward a new idea and plan for the further study of number theory,and will be an important direction of mathematical progress.This new field contains not only the proof of Fermat’s last theorem,but also many famous conjectures,such as Riemann hypothesis,Ramanujan conjecture,Lindelof conjecture and so on.This thesis mainly focuses on the Lindelof conjecture.Firstly,let f be a self-dual Hecke-Maass eigenform for the group SL3(Z).For 1/2<σ<1 fixed we define m(σ)≥2)as the supremum of all numbers m such that∫1T|L(σ+it,f)|mdt<<f,εT1+ε,where L(σ+it,f)is the Godement-Jacquet L-function related to f.In this thesis,we first show the lower bound of m(σ)for 2/3<σ<1<1.Then we establish asymptotic formulas for the second,fourth and sixth powers of L(σ+it,f)as applications.Secondly,suppose that π is a unitary cuspidal automorphic representation of GLr(AQ)and L(σ+it,π)is the automorphic L-function related to π.For 1/2<σ<1,let m(σ)≥2 be the supremum of all numbers m such that∫1T|L(σ+it,π)|mdt<<π,ε T1+ε.We are interested in the lower bound of m(σ)for 1-1/r<σ<1.Then as an application,we establish asymptotic formulas for the second,fourth and sixth power moments of L(σ+it,π).Finally,let f(z)be a Maass cusp form for the full modular group SL2(Z).And let λsym2f(n)be the n-th coefficient of symmetric square L-function associated with f(z).For A≥2 fixed we define M(A)as the infimum of all numbers M such that∫1T|L(1/2+it,sym2/)|Adt<<f TM+ε,where L(σ+it,sym2f)is the automorphic L-function attached to f.In this thesis,we will establish the upper bounds of M(A)and get the zero density estimates for L(σ+it,sym2f).In this thesis,we consider the problems of m(σ)and M(A)on GL3,and extend the problem of m(σ)to GL(r).