Third moment of the remainder term for Heisenberg manifolds

TL;DR

This paper establishes the asymptotic behavior of the third moment of the remainder term in Weyl's law for 3D Heisenberg manifolds, confirming its asymmetric distribution and supporting conjectures on its growth rate.

Contribution

It proves a precise third moment asymptotic formula for the remainder term in Weyl's law on Heisenberg manifolds, including explicit constants and generalizations to higher dimensions.

Findings

Third moment integral behaves as d_3 T^(13/4) with explicit d_3

Confirms asymmetric distribution of R(t) about the t-axis

Supports conjecture R(t)=O(t^(3/4+δ))

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.