Cramér's formula for Heisenberg manifolds (Q818294)

The authors consider the difference \(R(t)\) between the counting function \(N(t)\), representing the number of eigenvalues of the Laplace Beltrami operator on a \(3\)-dimensional Heisenberg manifold which are less or equal to \(t\), and the first term of the asymptotic expansion of \(N(t)\) as \(t\) tends to \(\infty\), and prove a formula for the asymptotics of \(\int_{1}^{T}| R(t)| ^{2}\,dt\) for \(T\) large. Then a similar result is proved for Heisenberg manifolds of dimension \(2n+1\) with \(n>1\).