Subconvexity Bounds of L-Functions

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。“这个建议的目的是打破凸性界限的$L$-函数的高秩群。主要研究者已经在突破GL（3）上自对偶L-函数的凸性界方面取得了第一个进展。本文从多个方面讨论了高秩群上L-函数凸性界的破缺问题。具体的结构，可能的困难进行了讨论，新的技术和工具被提及。这些项目是密切相关的一个最基本的开放问题在数论-林德洛夫假设度n L-功能。次凸性界是Lindeloff假设的第一步，它主要研究在某些几何空间上构造的数论函数（含一个复参数的无穷和）的大小。这些函数的边界不仅因为其理论价值而且因为其广泛的应用而重要。例如，一个应用是更强的量子唯一遍历性猜想，它告诉我们某些几何空间上某些测度的收敛性以及这些测度的收敛速度。这是数学物理中一个非常重要的问题。其他应用程序将在未来被发现。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."This proposal is aimed at breaking the convexity bounds for $L$-functions on high rank groups. The principal investigator has already made the first progress on breaking the convexity bounds for self dual L-functions on GL(3). The problems related to breaking the convexity bounds for L-functions on high rank groups in various aspects are discussed in the proposal. Concrete conjectures are made, possible difficulties are discussed, new techniques and tools are mentioned. These projects are closely related to one of the most fundamental open problems in number theory - the Lindeloff hypothesis for degree n L-functions. The subconvexity bounds are the first steps toward the Lindeloff hypothesis.The proposal is concerned with the study of the size of certain number theoretical functions (infinite sums with one complex parameter) which are constructed on some geometric spaces. Bounding such functions are important not only because of its theoretical value but also because of its vast applications. For example, one application would be a stronger quantum unique ergodicity conjecture which tells us the convergence of certain measures on certain geometric spaces as well as the rate of convergence of these measures. This is a very important problem in mathematical physics. Other applications will be found in the future.