On A Mixed Integral Moment Of L-functions

In the history of analytic number theory,the research of Riemann ζ-function and L-functions hold an extremely important position,and mathematicians derive a variety of theorems and conjectures in this area,such as the mean-value estimate of L-functions on the central line:(?)Among the above formula,φ represents an automorphic form,k is a positive integer which is greater than or equal to 1,T is an arbitrary positive number.Nowadays,the integral mean-value estimate of twisted L-functions is drawing attention,let f be the Hecke-Maass cusp form for SL3(Z),the definition of twisted L-function is as follows:(?)X is the Dirichlet character,and A(1,n)is the Fourier-Whittaker coefficient of f.Frot[5]had done some effective work on the discrete mean-value estimate of the product of twisted L-function L(s,fχ)and Dirichlet L-function L(s,χ)under the supposed condition (?)The main theorem this paper works on is similar with the above result,calculating the upper bound of the mixed integral mean-value of the product of L-function attached to the Hecke-Maass cusp form for SL3(2)and the Riemann ζ-function on the central line.Theorem Let f be a normalized Hecke-Maass cusp form on SL3(Z),L(s,f)is the L-function associated to it,ζ(s)is Riemann ζ-function.Let V be a smooth function which satisfies the decay condition V(j)(u)<<j 1 and Supp(V)(?)[1,2],for all ε>0,we have:(?)The estimate of I can be taken into many applications in analytic number theory.The location of zeros,the upper bounds and lower bounds for the size of L-functions have a close relationship with it.In fact,if we suppose the General Lindeloff Conjecture is true,we could directly get the result:I<<T1+(?).So the main work of this paper is very meaningful.The main method is using the approximate functional equations of L(f,s)and ζ(s),we will at first show the trivial bound,transforming the estimation to two parts of finite sum,then estimate them separately to get it.Then we use the staionary phase method of exponential function integral to improve the trivial upper bound to a non-trivial upper bound which is line with the result under General Lindelof Conjecture.