The Remainder in Weyl's Law for Heisenberg Manifolds

Abstract

We prove that the error term $R(\lambda)$ in Weyl's law is $\mathcal O_{\epsilon}(\lambda^{5/6 + \epsilon})$ for certain three-dimensional Heisenberg manifolds. We also show that the $L^2$-norm of the Weyl error term integrated over the moduli space of left-invariant Heisenberg metrics is $\ll \lambda^{3/4 + \epsilon}$. We conjecture that $R(\lambda) = \mathcal{O}_{\epsilon}(\lambda^{3/4 + \epsilon})$ is a sharp deterministic upper bound for Heisenberg three-manifolds.