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

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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.