对称平方L-函数的均值估计及其应用

自守L-函数是当今解析数论的重要研究内容之一．该项目以对称平方L-函数为研究对象，考虑其对应的均值估计及其应用．此问题与对称平方L-函数的广义Lindel?f猜想、亚凸性估计有着重要而深刻的联系． 申请人已经在对称平方L-函数的二次积分均值及其Fourier系数估计方面有了一定的工作基础．在项目实施过程中，我们期望借助于L-函数的渐近函数方程、Petersson迹公式、Kuznetsov迹公式、Kloosterman和的均值估计等工具来探讨如下两个问题：1)对称平方L-函数在特殊点1/2处的二次离散均值；2)对称平方L-函数在临界线上的四次积分均值估计．本项目预期的研究结果将对解决对称平方L-函数均值意义下的广义Lindel?f猜想产生重要影响．

Automorphic L-function is one of the central topics nowadays in the area of analytic number theory. We aim at investigating the moments of the symmetric square L-function and their applications. The topic is quite important in the theory of L-functions, due to its connection with the generalized Lindel?f Hypothesis and the subconvexity of the symmetric square L-functions. For the symmetric square L-functions, the applicant has the working foundations for the estimation of Fourier coeffcients and the mean values. Using the approximate functional equation of L-function, Petersson trace formula, Kuznetsov trace formula and the average estimate of the Kloosterman sum , we are mainly concerned with the following two problems: 1) the second discrete moment for the symmetric square L-functions at the point 1/2； 2) the fourth integral moment for the symmetric square L-functions on the critical line. The expected results will contribute to solving the generalized Lindel?f Hypothesis for the symmetric square L-functions on average.