Skip to main content
SpringerLink
Log in

On the error term in Weyl’s law for the Heisenberg manifolds (II)

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper we study the mean square of the error term in the Weyl’s law of an irrational (2l + 1)-dimensional Heisenberg manifold. An asymptotic formula is established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hörmander L. The spectral function of an elliptic operator. Acta Math, 121:193–218 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bentkus V, Götze F. Lattice point problems and distribution of values of quadratic forms. Ann of Math, 50(2):977–1027 (1999)

    Article  Google Scholar 

  3. Bérard P H. On the wave equation on a compact Riemannian manifold without conjugate points. Math Z, 155:249–276 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bleher L. On the distribution of the number of lattice points inside a family of convex ovals. Duke Math J, 67:461–481 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chung D, Petridis Y N, Toth J. The remainder in Weyl’s law for Heisenberg manifolds II. Bonner Mathematische Schriften, Nr 360, Bonn, 2003

  6. Fricker F. Einführung in die Gitterpunketlehre [Introduction to lattice point theory]. Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften(LMW), Mathematische Reihe[Textbooks and Monographs in the Exact Sciences], Basel-Boston, Mas: Birkhäuser Verlag, 1982, 73

    Google Scholar 

  7. Götze F. Lattice point problems and values of quadratic forms. Invent Math, 157:195–226 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Huxley M N. Exponential sums and lattice points III. Proc London Math Soc, 87(3):591–609 (2003)

    Article  MathSciNet  Google Scholar 

  9. Ivrii V YA. Precise Spectral Asymptotics for elliptic Operators Acting in Fibrings over Manifolds with Boundary. Lecture Notes in Mathematics, Vol 1100, Berlin: Springer, 1984

    Google Scholar 

  10. Khosravi M, Petridis Y. The remainder in Weyl’s law for n-dimensional Heisenberg manifolds. Proc Amer Math Soc, 133:3561–3571 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Petribis Y, Toth J. The remainder in Weyl’s law for Heisenberg manifolds. J Differential Geom, 60:455–483 (2002)

    MathSciNet  Google Scholar 

  12. Volovoy A V. Improved two-term asymptotics for the eigenvalue distribution function of an elliptic operator on a compact manifold. Comm Partial Differemtial Equations, 15:1509–1563 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hardy G H. On the expression of a number as the sum of two squares. Quart J Math, 46:263–283 (1915)

    MATH  Google Scholar 

  14. Kátai I. The number of lattice points in a circle (in Russian). Ann Univ Sci Budapest Rolando Eötvös, Sect Math, 8:39–60 (1965)

    Google Scholar 

  15. Tsang K M. Higher-power moments of Δ(x), E(t) and P(x). Proc London Math Soc, 65:65–84 (1992)

    Article  MathSciNet  Google Scholar 

  16. Zhai W G. On higher-power moments of Δ(x) (II). Acta Arith, 114:35–54 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhai W G. On higher-power moments of Δ(x) (III). Acta Arith, 118:263–281 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Khosravi M, Toth J. Cramér’s formula for Heisenberg manifolds. Ann de l’institut Fourier, 55:2489–2520 (2005)

    MATH  MathSciNet  Google Scholar 

  19. Khosravi M. Third moment of the remainder error term in Weyl’s law for Heisenberg manifolds. ArXiv: 0711-0073

  20. Zhai W G. On the error term in Weyl’s law for the Heisenberg manifolds. Acta Arith, 134:219–257 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Roth K F. Rational aproximation to algebraic numbers. Mathematica, 2:1–20 (1955)

    Google Scholar 

  22. Khinchin A Y. Zur metrischen Theorie der diophantischen approximationen. Math Z, 24:706–714 (1926)

    Article  MathSciNet  Google Scholar 

  23. Folland G B. Harmonic Analysis in Phase Space. Princeton: Princeton University Press, 1989, 9–73

  24. Gordon C, Wilson E. The spectrum of the Laplacian on Riemannian Heisenberg manifolds. Michigan Math J, 33:253–271 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stein E M. Harmonic Analysis. Princeton: Princeston University Press, 1993, 527–574

    MATH  Google Scholar 

  26. Vaaler J D. Some extremal functions in Fourier analysis. Bull Amer Math Soc, 12:183–216 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  27. Heath-Brown D R. The Piatetski-Shapiro prime theorem. J Number Theory, 16:242–266 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  28. Min S H. The Methods of Number Theory (in Chinese). Beijing: Science Press, 1981

    Google Scholar 

  29. Vinogradov I M. Special Variants of The Method of Trigonometric Sums. Moscow: Nauka, 1976; English transl in his Selected works. New York: Springer-Verlag, 1985

    MATH  Google Scholar 

  30. Kuhleitner M, Nowak W G. The asymptotic behaviour of the mean-square of fractional part sums. Proc Edinb Math Soc, 43:309–323 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Shandong Normal University, Jinan, 250014, China

    WenGuang Zhai

Authors
  1. WenGuang Zhai
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This work was supported by National Natural Science Foundation of China (Grant No. 10771127)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhai, W. On the error term in Weyl’s law for the Heisenberg manifolds (II). Sci. China Ser. A-Math. 52, 857–874 (2009). https://doi.org/10.1007/s11425-008-0167-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-008-0167-z

Keywords

MSC(2000)

Search

Navigation