自守形式和自守L-函数是数论中神秘且特别常见的研究对象和研究工具，而Sarnak提出的Möbius正交性猜想是数论及其相关领域中的重要研究主题。本项目拟研究高阶自守形式情形的Möbius正交性猜想，具体是指Iwaniec、Luo和Sarnak提出的“假设C”或者Titchmarsh除数问题的高阶情形，我们期望证明其强正交性或有指数结余。进而，申请人拟用高阶自守形式情形的Möbius正交性猜想结合Duke-Iwaniec方法来研究L-函数在临界带内的非零性，期望获得在扭乘Dirichlet特征L-函数的族类中有正比例的数量是非零的，并且进一步应用到高阶群上的Rankin-Selberg L-函数和对称幂L-函数。申请人研究方向是自守形式和L-函数，在此方向已做出重要工作，且正式发表了十篇Sci论文，这些都确保了本项目能够顺利完成。

Automorphic forms and automorphic L-functions appear to be esoteric and special topics or tools in number theory, and Sarnak's Möbius disajointness conjecture is an important theme in number theory and related areas. This project concerns the Möbius disjointness conjecture on automorphic forms, especially the high rank cases of “Hypothesis C” proposed by Iwaniec-Luo-Sarnak or Titchmarsh's divisor problem. The investigator expects to prove some related results with the strong orthogonality or the power savings. Moreover, together with the approachs of Duke and Iwaniec, the investigator intends to consider the non-vanishing of L-functions in the critical strips and expects to obtain some non-vanishing results for a positive proportion of twisted L-functions, which further have applications to the Rankin-Selberg L-functions and symmetric power L-functions on higher rank groups. The research areas of the investigator are automorphic forms and L-functions. The investigator has shown some important results and published about ten papers. These all ensure the successful accomplishment of this project.