Third moment of the remainder term for Heisenberg manifolds

Abstract

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_\delta(T^(45/14+\delta)), 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_\delta(t^(3/4+\delta)). Similar results hold for (2n+1)-dimensional Heisenberg manifolds with arithmetic metrics.