论文标题
兰金·塞尔伯格（Rankin-Selberg
First moments of Rankin-Selberg convolution of Automorphic Forms on $GL(2)$
论文作者
论文摘要
我们获得了Rankin-Selberg卷积$ L $ l $ l $ l $ l $ lomorphic模块化形式或MAASS在$ gl（2）$上的任意水平的第一刻公式，并具有Maass形式的正顺序。结果之一是迄今为止的最佳结果，水平，频谱价值和重量均匀，对于两个Maass或Holomorphic cusp的平等，如果其兰金·塞尔伯格（Rankin-Selberg）的兰金·塞尔伯格（Rankin-Selberg）汇合质量$ u_j $的正常基础$ u_j $在关键带的中心是足够多的$ u_j $。 我们方法的主要新颖性是对错误项的处理方式。它们被带入确切的形式，为第一个时刻案例提供了最佳估计，还为将其扩展到第二时刻提供了基础，这将出现在另一项工作中。
We obtain a first moment formula for Rankin-Selberg convolution $L$-series of holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an orthonormal basis of Maass forms. One consequence is the best result to date, uniform in level, spectral value and weight, for the equality of two Maass or holomorphic cusp forms if their Rankin-Selberg convolutions with the orthonormal basis of Mass forms $u_j$ is equal at the center of the critical strip for sufficiently many $u_j$. The main novelty of our approach is the new way the error terms are treated. They are brought into an exact form that provides optimal estimates for the first moment case, and also provide a basis for an extension to second moments, which will appear in another work.
