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

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.

References

V. Bentkus and F. Götze, Lattice point problems and distribution of values of quadratic forms, Ann. of Math. (2) 150 (1999), no. 3, 977–1027. MR 1740988, DOI 10.2307/121060
Pierre H. Bérard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), no. 3, 249–276. MR 455055, DOI 10.1007/BF02028444
Pavel Bleher, On the distribution of the number of lattice points inside a family of convex ovals, Duke Math. J. 67 (1992), no. 3, 461–481. MR 1181309, DOI 10.1215/S0012-7094-92-06718-4
Derrick Chung, Yiannis N. Petridis, and John A. Toth, The remainder in Weyl’s law for Heisenberg manifolds. II, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, vol. 360, Univ. Bonn, Bonn, 2003, pp. 16. MR 2075620
Harald Cramér, Über zwei Sätze des Herrn G. H. Hardy, Math. Z. 15 (1922), no. 1, 201–210 (German). MR 1544568, DOI 10.1007/BF01494394
Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
François Fricker, Einführung in die Gitterpunktlehre, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften (LMW). Mathematische Reihe [Textbooks and Monographs in the Exact Sciences. Mathematical Series], vol. 73, Birkhäuser Verlag, Basel-Boston, Mass., 1982 (German). MR 673938
A. Good, Ein $\Omega$-Resultat für das quadratische Mittel der Riemannschen Zetafunktion auf der kritischen Linie, Invent. Math. 41 (1977), no. 3, 233–251 (German). MR 460253, DOI 10.1007/BF01403049
Carolyn S. Gordon and Edward N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), no. 2, 253–271. MR 837583, DOI 10.1307/mmj/1029003354
Friedrich Götze, Lattice point problems and values of quadratic forms, Invent. Math. 157 (2004), no. 1, 195–226. MR 2135188, DOI 10.1007/s00222-004-0366-3
G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J Math. 46(1915), 263-283.
D. R. Heath-Brown and K. Tsang, Sign changes of $E(T)$, $\Delta (x)$, and $P(x)$, J. Number Theory 49 (1994), no. 1, 73–83. MR 1295953, DOI 10.1006/jnth.1994.1081
Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 609014, DOI 10.1007/BF02391913
M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc. (3) 87 (2003), no. 3, 591–609. MR 2005876, DOI 10.1112/S0024611503014485
Aleksandar Ivić, The Riemann zeta-function, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1985. The theory of the Riemann zeta-function with applications. MR 792089
A. Ivić and Wenguang Zhai, Higher moments of the error term in the divisor problem, to appear in Mat. Zametki.
Victor Ivriĭ, Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary, Lecture Notes in Mathematics, vol. 1100, Springer-Verlag, Berlin, 1984. MR 771297, DOI 10.1007/BFb0072205
Matti Jutila, On the divisor problem for short intervals, Ann. Univ. Turku. Ser. A I 186 (1984), 23–30. Studies in honour of Arto Kustaa Salomaa on the occasion of his fiftieth birthday. MR 748516
M. Khosravi, Third moment of the remainder error term in Weyl’s law for Heisenberg manifolds, arXiv: 0711.0073.
Mahta Khosravi and Yiannis N. Petridis, The remainder in Weyl’s law for $n$-dimensional Heisenberg manifolds, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3561–3571. MR 2163591, DOI 10.1090/S0002-9939-05-08155-4
Mahta Khosravi and John A. Toth, Cramér’s formula for Heisenberg manifolds, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 7, 2489–2520 (English, with English and French summaries). MR 2207391
Yuk-Kam Lau and Kai-Man Tsang, On the mean square formula of the error term in the Dirichlet divisor problem, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 277–287. MR 2475967, DOI 10.1017/S0305004108001874
W. G. Nowak, A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds, arXiv: 0809.3924.
W. G. Nowak, A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds.II, arXiv: 0810.2235.
Yiannis N. Petridis and John A. Toth, The remainder in Weyl’s law for Heisenberg manifolds, J. Differential Geom. 60 (2002), no. 3, 455–483. MR 1950173
K. Soundararajan, Omega results for the divisor and circle problems, Int. Math. Res. Not. 36 (2003), 1987–1998. MR 1991181, DOI 10.1155/S1073792803130309
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Kai Man Tsang, Higher-power moments of $\Delta (x),\;E(t)$ and $P(x)$, Proc. London Math. Soc. (3) 65 (1992), no. 1, 65–84. MR 1162488, DOI 10.1112/plms/s3-65.1.65
A. V. Volovoy, Improved two-term asymptotics for the eigenvalue distribution function of an elliptic operator on a compact manifold, Comm. Partial Differential Equations 15 (1990), no. 11, 1509–1563. MR 1079602, DOI 10.1080/03605309908820736
Wenguang Zhai, On higher-power moments of $\Delta (x)$. II, Acta Arith. 114 (2004), no. 1, 35–54. MR 2067871, DOI 10.4064/aa114-1-3
Wenguang Zhai, On higher-power moments of $\Delta (x)$. III, Acta Arith. 118 (2005), no. 3, 263–281. MR 2168766, DOI 10.4064/aa118-3-3
Wenguang Zhai, On the error term in Weyl’s law for Heisenberg manifolds, Acta Arith. 134 (2008), no. 3, 219–257. MR 2438847, DOI 10.4064/aa134-3-3