Research on density theorems and invariants with zeta functions and trace formulas

ζ函数和密度定理之间有很深的联系，比如黎曼的ζ函数和素数定理，塞尔伯格的ζ函数和素数测地线定理。在本研究中，我们的目的是通过研究密度定理和不变量与ζ函数和迹公式来解释流形的性质。特别是由双曲流形上的封闭测地线长度集合所定义的长度谱，对于表征该流形是非常重要的。本研究的主要成果是用经典数论中的对象来描述算术曲面的长度谱，并得到了一个解释长度谱多重性的“平均”行为的渐近公式。作为应用，我们也得到了不定二元二次型类数渐近公式的一种改进。

There are deep connections between zeta functions and density theorems, such like Riemann's zeta functions and the prime number theorem, Selberg's zeta functions and the prime geodesic theorem. In this research, we aim to explain the properties of manifolds by studying density theorems and invariants with zeta functions and trace formulas. Especially, the length spectrum defined by the set of length of closed geodesics on a hyperbolic manifold is important to characterize the manifold. The main results on this research is to describe the length spectra for arithmetic surfaces in terms of objects in the classical number theory, and to obtain an asymptotic formula to explain ``average" of the behavior of the multiplicity of the length spectrum. As an application, we also get one kind of improvements of the asymptotic formula for the class numbers of indefinite binary quadratic forms.