ANNALES DE L'INSTITUT FOURIER

Cramér's formula for Heisenberg manifolds
Annales de l'Institut Fourier, Volume 55 (2005) no. 7, p. 2489-2520
Let $R\left(\lambda \right)$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that ${\int }_{1}^{T}{|R\left(t\right)|}^{2}dt=c{T}^{\frac{5}{2}}+{O}_{\delta }\left({T}^{\frac{9}{4}+\delta }\right)$, 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.
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 ${\int }_{1}^{T}{|R\left(t\right)|}^{2}dt=c{T}^{\frac{5}{2}}+{O}_{\delta }\left({T}^{\frac{9}{4}+\delta }\right)$, 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 $2n+1$.
DOI : https://doi.org/10.5802/aif.2168
Classification:  35P20,  58J50
Keywords: Heisenberg manifolds, Weyl's law, Cramér's formula, poisson summation formula
Keywords: Heisenberg manifolds, Weyl's law, Cramér's formula, poisson summation formula
@article{AIF_2005__55_7_2489_0,
author = {Khosravi, Mahta and Toth, John A.},
title = {Cram\'er's formula for Heisenberg manifolds},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {55},
number = {7},
year = {2005},
pages = {2489-2520},
doi = {10.5802/aif.2168},
mrnumber = {2207391},
zbl = {1090.58018},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_2005__55_7_2489_0}
}
Cramér's formula for Heisenberg manifolds. Annales de l'Institut Fourier, Volume 55 (2005) no. 7, pp. 2489-2520. doi : 10.5802/aif.2168. https://aif.centre-mersenne.org/item/AIF_2005__55_7_2489_0/

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