2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141219Let 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.16 pagesAnalysis of PDEsSpectral Theorytext