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Cramér's formula for Heisenberg manifolds

Mahta KhosraviJohn A. Toth — 2005

Annales de l'institut Fourier

Let R ( λ ) be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that 1 T | R ( t ) | 2 d t = c T 5 2 + O δ ( T 9 4 + δ ) , where c is a specific nonzero constant and δ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that R ( t ) = O δ ( t 3 4 + δ ) .The idea of the proof is to use the Poisson summation formula to write the error term in a form which can be estimated by the method of the stationary phase. The similar result will be also proven in the 2 n + 1 -dimensional case.

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