Moments And Upper Bounds For L-functions In The Weight Aspect

The problem of estimating the mean value of automorphic L-functions is one of the core problems in analytic number theory.In this paper,we present a new proof of the mean value estimate and upper bound estimate for GL(2)Lfunctions in the weight aspect,starting from the Rankin-Selberg convolution of GL(3)Eisenstein series and GL(2)Eisenstein series.More specifically,let {uj} be an orthonormal basis of even Hecke-Maass forms for SL2(Z)with the Laplacian eigenvalue 1/4+tj2 with tj≥ 0.Then for ε>0,large T and Tε≤M≤T1-ε,we have where ’ means summing over the orthonormal basis of even Hecke-Maass forms.The main innovation of this paper is the treatment for the integral on the right-hand side of the inverse Voronoi formula.We combine the integral with the exponential factor and provide its asymptotic formula.This treatment not only reduces the oscillation of the integral but also employs the large sieve inequality to provide an effective structure.Thus,we establish the upper bound of the mean value of the L-functions.