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

Tsang, Kai-Man; Zhai, Wenguang

doi:10.1090/S0002-9947-2012-05437-7

Sign changes of the error term in Weyl’s law for Heisenberg manifolds
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by Kai-Man Tsang and Wenguang Zhai

Trans. Amer. Math. Soc. 364 (2012), 2647-2666

DOI: https://doi.org/10.1090/S0002-9947-2012-05437-7

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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