Heisenberg流形余项的三阶矩
Third moment of the remainder term for Heisenberg manifolds
摘要：
设R（t）是具有一定算术度量的三维黎曼海森堡流形的外尔定律的余项。证明了一个三阶矩结果：\int_1^T R（t）^3 dt =d_3 T^（13/4）+O_δ（T^（45/14+δ）），其中d_3是一个可以显式计算的常数.这证明了R（t）关于t轴的不对称行为。这个结果与Petridis和Toth的猜想R（t）=O_δ（t^（3/4+δ））是一致的.对于具有算术度量的（2n+1）维海森堡流形，也有类似的结果.
英文摘要：
Let R(t) be the remainder term in Weyl's law for a 3-dimensional Riemannian Heisenberg manifold with a certain arithmetic metric. We prove a third moment result stating that \int_1^T R(t)^3 dt =d_3 T^(13/4)+O_δ(T^(45/14+δ)), where d_3 is a specific positive constant which can be evaluated explicitly. This proves the asymmetric behavior of R(t) about the t-axis. This result is consistent with the conjecture of Petridis and Toth stating that R(t)=O_δ(t^(3/4+δ)). Similar results hold for (2n+1)-dimensional Heisenberg manifolds with arithmetic metrics.
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