Cramér's formula for Heisenberg manifolds

Khosravi, Mahta; Toth, John A.

Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2489-2520.

Soit $R\left(\lambda \right)$ le terme d’erreur de la loi de Weyl pour une variété riemannienne de Heisenberg de dimension 3. Nous prouvons que ${\int }_1^T{\left|R\left(t\right)\right|}^{2}dt=c{T}^{\frac{5}{2}}+O_{\delta }\left(T^{\frac{9}{4}+\delta }\right)$, où $c$ est une constante spécifique non nulle et $\delta$ est un nombre positif arbitrairement petit. Ce résultat est une avancée vers la conjecture de Petridis et Toth, qui énonce que $R\left(t\right)=O_{\delta }\left(t^{\frac{3}{4}+\delta }\right)$. L’idée de la preuve est d’utiliser la formule de sommation poisson pour réécrire le terme d’erreur sous une forme qui est majorable au moyen de la méthode des phases stationnaires. Le même résultat sera prouvé pour la dimension $2n+1$.

Let $R\left(\lambda \right)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that ${\int }_1^T{\left|R\left(t\right)\right|}^{2}dt=c{T}^{\frac{5}{2}}+O_{\delta }\left(T^{\frac{9}{4}+\delta }\right)$, where $c$ is a specific nonzero constant and $\delta$ is an arbitrary small positive number. This is consistent with the conjecture of Petridis and Toth stating that $R\left(t\right)=O_{\delta }\left(t^{\frac{3}{4}+\delta }\right)$.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 $2n+1$-dimensional case.

Classification : 35P20, 58J50
Keywords: Heisenberg manifolds, Weyl's law, Cramér's formula, poisson summation formula, Heisenberg manifolds, Weyl's law, Cramér's formula, poisson summation formula
Mot clés : variété d'Heisenberg, loi de Weyl, formule de Cramér, formule de sommation de Poisson

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