The Mean Value Of The Riesz Mean Error Term Of Symmetric Square L-function

Langlands Program closely links algebra,number theory,harmonic analysis and geometry.Most of forward subjects in modern number theory are carried out around Langlands Program.The automorphic form theory is the core content of Langlands Program.It is an emerging field in the development of mathematics and provides new ideas and directions for the study of number theory.The work of this thesis will focus on the automorphic function in the automorphic form theory.The Dirichlet divisor problem D(x)=(?)d(n)is a classical problem in analytic number theory,where the Dirichlet divisor function d(n)=(?)1(n,n1 and n2 are all positive integers)is an number-theoretic function,and d(n)is the coefficient of ζ2(s).It is well known that the asymptotic formula of D(x)is of the form D(x)=x(logx+2γ-1)+Δ(x),where γ=0.57721...is the Euler’s constant and Δ(x)is the error term.Therefore,the goal of the Dirichlet divisor problem is to improve the size of the error term Δ(x).Many scholars have studied the upper bound,lower bound and mean value estimation of the integral of Δ(x)and obtained a series of results.This thesis mainly studies the similar problems of the automorphic functions.Denote Sl(Γ)the space of holomorphic cusp forms of even integral weight k for the full modular group Γ=SL(2,Z).f(z)is an eigenfunction of all the Hecke operators belonging to Sl(Γ).The Hecke L-function attached to f(z)is defined for(?)(s)>1 by L(s,f)=(?)λf(n)n-s.The mean value of the coefficient λf(n)is denoted by Sf(x).The symmetric square L-function attached to f(z)is defined as L(s,sym2f)=(?)cnn-s for(?)(s)＞1.The error term of the Riesz mean value of the coefficient cn is denoted by Δρ(x;sym2f)(0≤ρ≤1).In particular,when ρ=0 is recorded as Ssym2f(x).This thesis mainly studies the higher power moments of Δρ(x;sym2f).Firstly,the asymptotic formula of (?)is established in the short interval[T-H,T+H](T(?)≤H≤T)with k≥ 3.Secondly,when the integral interval is extended to[1,T]and k=3,4,5,the asymptotic formula of (?) is obtained.Finally,some new hybrid moment results of Sf(x)and Ssym2f(x)are established in the interval[1,T].