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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sign changes of the error term in Weyl's law for Heisenberg manifolds


Authors: Kai-Man Tsang and Wenguang Zhai
Journal: Trans. Amer. Math. Soc. 364 (2012), 2647-2666
MSC (2010): Primary 11N37, 35P20, 58J50
Published electronically: January 19, 2012
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Abstract: Let $ R(T)$ be the error term in Weyl's law for the $ (2l+1)$-dimen-
sional Heisenberg manifold $ (H_l/\Gamma , g_l)$. In this paper, several results on the sign changes and odd moments of $ R(t)$ are proved. In particular, it is proved that for some sufficiently large constant $ c$, $ R(t)$ changes sign in the interval $ [T, T + c \sqrt T]$ for all large $ T$. Moreover, for a small constant $ c_1$ there exist infinitely many subintervals in $ [T, 2T]$ of length $ c_1 \sqrt T \log ^{-5} T$ such that $ \pm R(t)>c_1t^{l - 1/4}$ holds on each of these subintervals.


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Additional Information

Kai-Man Tsang
Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam road, Hong Kong
Email: kmtsang@maths.hku.hk

Wenguang Zhai
Affiliation: Department of Mathematics, China University of Mining and Technology, Beijing 100083, People’s Republic of China
Email: zhaiwg@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05437-7
PII: S 0002-9947(2012)05437-7
Keywords: Heisenberg manifold, Weyl’s law, error term, Voronoi’s formula, sign change.
Received by editor(s): December 5, 2009
Received by editor(s) in revised form: July 24, 2010
Published electronically: January 19, 2012
Additional Notes: The work of the second author was supported by National Natural Science Foundation of China (Grant No. 10771127) and Mathematical Tianyuan Foundation (No. 10826028).
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.