# Cramér's formula for Heisenberg manifolds

Mahta Khosravi[1]; John A. Toth[1]

• [1] McGill University, Department of Mathematics and Statistics, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6 (Canada)
• Volume: 55, Issue: 7, page 2489-2520
• ISSN: 0373-0956

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## Abstract

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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.

## How to cite

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Khosravi, Mahta, and Toth, John A.. "Cramér's formula for Heisenberg manifolds." Annales de l'institut Fourier 55.7 (2005): 2489-2520. <http://eudml.org/doc/116261>.

@article{Khosravi2005,
abstract = {Let $R(\lambda )$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that $\int _1^T \vert R(t)\vert ^2 dt =c T^\{\{5\} \over \{2\}\}+O_\{\delta \}(T^\{\{\{9\} \over \{4\}\}+\delta \})$, 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 $R(t)=O_\{\delta \}(t^\{\{\{3\} \over \{4\}\}+\delta \})$.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.},
affiliation = {McGill University, Department of Mathematics and Statistics, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6 (Canada); McGill University, Department of Mathematics and Statistics, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6 (Canada)},
author = {Khosravi, Mahta, Toth, John A.},
journal = {Annales de l'institut Fourier},
keywords = {Heisenberg manifolds; Weyl's law; Cramér's formula; poisson summation formula; Cramér’s formula; Poisson summation formula},
language = {eng},
number = {7},
pages = {2489-2520},
publisher = {Association des Annales de l'Institut Fourier},
title = {Cramér's formula for Heisenberg manifolds},
url = {http://eudml.org/doc/116261},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Khosravi, Mahta
AU - Toth, John A.
TI - Cramér's formula for Heisenberg manifolds
JO - Annales de l'institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 7
SP - 2489
EP - 2520
AB - Let $R(\lambda )$ be the error term in Weyl’s law for a 3-dimensional Riemannian Heisenberg manifold. We prove that $\int _1^T \vert R(t)\vert ^2 dt =c T^{{5} \over {2}}+O_{\delta }(T^{{{9} \over {4}}+\delta })$, 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 $R(t)=O_{\delta }(t^{{{3} \over {4}}+\delta })$.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.
LA - eng
KW - Heisenberg manifolds; Weyl's law; Cramér's formula; poisson summation formula; Cramér’s formula; Poisson summation formula
UR - http://eudml.org/doc/116261
ER -

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