关于Beurling-Wintner问题的研究

本项目研究基于Beurling-Wintner问题等一系列重要问题展开。该问题是由两位数学家A. Beurling和A. Wintner独立提出的经典分析难题，它与Riemann猜想密切联系，在解析数论中具有重要的理论意义。H. Hedenmalm，P. Lindqvist和K. Seip在解析Hilbert空间的框架下系统性地研究了该问题，并证明了它等价于无限维多圆盘Hardy空间的循环向量问题。N. Nikolski在这些问题尤其是循环向量问题上作出了重要的贡献。最近，关于带有理间断点的阶梯函数类，我们利用解析数论方法完全解决了Beurling-Wintner问题。特别地，我们完全解决了Kozlov完全性问题的有理数版本。本项目将在上述研究工作的基础上，进一步在这些重要问题上取得本质进展。此外，本项目将尝试融入解析数论、遍历论、动力系统中的方法，为算子理论的研究开辟新的思路。

This project is based on a series of important problems, including the Beurling-Wintner problem. This problem is a classical analysis problem raised independently by mathematicians A. Beurling and A. Wintner. It is closely related to the Riemann Hypothesis, and of theoretical significance in analytic number theory. H. Hedenmalm, P. Lindqvist and K. Seip studied systematically the problem in the frame of analytic Hilbert spaces. Also, they proved that the problem is equivalent to the cyclic vector problem on the Hardy space over the infinite-dimensional polydisk. N. Nikolski made significant contributions to these problem, especially the cyclic vector problem. Recently, we completely solve the Beurling-Wintner problem for step functions with rational jump discontinuities by approaches from analytic number theory. This leads to a complete solution to the rational version of the Kozlov completeness problem. On the basis of the above researches, this project will make essential progresses on these important problems. Moreover, the project will involve approaches from analytic number theory, ergodic theory, dynamical system, which could open a new door for researches in operator theory.