GL(n)上的Hecke-Maass尖形式是Langlands纲领中的重要内容，它们对应的自守L-函数由它们的Hecke特征值定义．因此，对它们的Hecke特征值的研究具有非常重要的理论意义．本项目拟主要研究它们的Hecke特征值的分布，包括以下四个课题：.(1)每个素数p对应的GL(n)上的“例外”Hecke-Maass尖形式的个数的上界估计；.(2)GL(n)上的Hecke-Maass尖形式在平均意义下的Sato-Tate猜想及其残差分析；.(3)GL(n)上的Hecke-Maass尖形式的垂直Sato-Tate猜想的收敛速度；.(4)GL(n)上的Hecke-Maass尖形式的Elliott-Montgomery-Vaughan类型的大筛法型.不等式及其应用...本项目预期的研究结果将会对GL(n)上的Hecke特征值的研究产生重要影响.

Hecke-Maass cusp forms play an important role in Langlands Program. Their corresponding automorphic L-functions are defined by their Hecke eigenvalues. Therefore, it is of great theoretical significance to investigate their Hecke eigenvalues. In this project, we focus on the distribution of their Hecke eigenvalues and study the following four topics:.(1) The number of "exceptional" Hecke-Maass cusp forms on GL(n);.(2) Fluctuations in the Sato-Tate conjecture on average for GL(n);.(3) Quantitative version of the vertical Sato-Tate conjecture for GL(n);.(4) A large sieve inequality of Elliott-Montgomery-Vaughan type for .Hecke-Maass cusp forms on GL(n) with applications...The expected results of this project will have important impacts on the study of Hecke eigenvalues for GL(n).