Average behavior of the triple divisor function over values of quadratic form

Abstract

In this paper, we apply the circle method to study the average behavior of the triple divisor function over values of quadratic form \( {n}_1^2+{n}_2^2+\cdots +{n}_l^2 \) with l ⩾ 3. We improve and generalize previous results.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    C. Calderón and M. J. de Velasco, On divisors of a quadratic form, Bol. Soc. Bras. Mat., Nova Sér., 31(1):81–91, 2000.

    MathSciNet  Article  Google Scholar 

  2. 2.

    F. Chamizo and H. Iwaniec, On the sphere problem, Rev. Mat. Iberoam., 11(2):417–429, 1995.

    MathSciNet  Article  Google Scholar 

  3. 3.

    J. Chen, Improvement of asymptotic formulas for the number of lattice points in a region of three dimensions. II, Sci. Sin., 12:751–764, 1963.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    R. T. Guo and W. G. Zhai, Some problems about the ternary quadratic form \( {m}_1^2+{m}_2^2+{m}_3^2 \), Acta Arith., 156(2):101–121, 2012.

    MathSciNet  Article  Google Scholar 

  5. 5.

    D. R. Heath-Brown, Lattice points in the sphere, in Number Theory in Progress, Vol. 2, Walter de Gruyter, Berlin, 1999, pp. 883–892.

    Google Scholar 

  6. 6.

    L. Q. Hu and L. Yang, Sums of the triple divisor function over values of a quaternary quadratic form, Acta Arith., 183(1):63–85, 2018.

    MathSciNet  Article  Google Scholar 

  7. 7.

    M. N. Huxley, Area, Lattice Points, and Exponential Sums, Lond. Math. Soc. Monogr., New Ser., No. 13, Clarendon Press, Oxford, 1996.

  8. 8.

    A. Ivimć, On the ternary additive divisor problem and the sixth moment of the zeta-function, in G. R. H. Greaves, G. Harman, and M. N. Huxley (Eds.), Sieve methods, Exponential Sums, and Their Applications in Number Theory. Proceedings of a Symposium Held in Cardiff, July 1995), Lond. Math. Soc. Lect. Note Ser., Vol. 237, Cambridge Univ. Press, Cambridge, 1997, pp. 205–243.

  9. 9.

    H. Iwaniec and E. Kowalski, Analytic Number Theory, Colloq. Publ., Am. Math. Soc., Vol. 53, AMS, Providence, RI, 2004.

  10. 10.

    Y. J. Jiang and G. S. Lü, Shifted convolution sums for higher rank groups, Forum Math., 31(2):361–383, 2019.

    MathSciNet  Article  Google Scholar 

  11. 11.

    X. Q. Li, The Voronoi formula for the triple divisor function, in J. Liu (Ed.), Lecture Notes from the Conference CIMPA-UNESCO-CHINA Research School 2010: Automorphic Forms and L-functions,Weihai, China, August 1–14, 2010, Adv. Lect. Math. (ALM), Vol. 30, Int. Press / Higher Education Press, Somerville, MA/ Beijing, 2014, pp. 69–90.

  12. 12.

    X. M. Ren and Y. B. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GLm(ℤ), Sci. China, Math., 58(10):2105–2124, 2015.

    MathSciNet  Article  Google Scholar 

  13. 13.

    P. Shiu, A Brun–Titchmarsh theorem for multiplicative functions, J. Reine Angew. Math., 313:161–170, 1980.

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Q. F. Sun and D. Y. Zhang, Sums of the triple divisor function over values of a ternary quadratic form, J. Number Theory, 168:215–246, 2016.

    MathSciNet  Article  Google Scholar 

  15. 15.

    R. Vaughan, The Hardy–Littlewood Method, 2nd ed., Cambridge Tracts Math., Vol. 125, Cambridge Univ. Press, Cambridge, 1997.

    Google Scholar 

  16. 16.

    I. M. Vinogradov, On the number of integer points in a sphere, Izv. Akad. Nauk SSSR, Ser. Mat., 27:957–968, 1963 (in Russian).

  17. 17.

    L. L. Zhao, The sum of divisors of a quadratic form, Acta Arith., 163(2):161–177, 2014.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Guangwei Hu.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported in part by NSFC (Nos. 11771252, 11531008).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hu, G., Hu, W. Average behavior of the triple divisor function over values of quadratic form. Lith Math J (2020). https://doi.org/10.1007/s10986-020-09500-x

Download citation

Keywords

  • triple divisor function
  • circle method

MSC

  • 11E25
  • 11P55