Advertisement

The Ramanujan Journal

, Volume 47, Issue 1, pp 1–19 | Cite as

A note on shifted convolution of cusp-forms with \(\theta \)-series

  • Xiaoguang He
  • Yujiao Jiang
Article

Abstract

In this paper, we study shifted convolution sums for GL(2) and give an upper bound for \(\sum \nolimits _{n\ge 1}\lambda _f(n+b) r(n,Q)g(n)\), where g(n) is a smooth weight function. In particular, we get an upper bound for \(\sum \nolimits _{n\le x}\lambda _f(n+b)r_3(n)\), which improves the result in Lü et al. (Ramanujan J 40(1):C115–C133, 2016).

Keywords

Shifted convolution sum Cusp forms Theta series 

Mathematics Subject Classification

11F30 11F11 11F27 11F37 

Notes

Acknowledgements

The authors are very grateful to the referee for thorough reading of the paper and many valuable suggestions that clarify the argument of the paper.

References

  1. 1.
    Bateman, P.T.: On the representations of a number as the sum of three squares. Trans. Am. Math. Soc. 71, 70–101 (1951)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Deligne, P.: La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Duke, W., Friedlander, B.J., Iwaniec, H.: Bounds for automorphic \(L\)-functions. III. Invent. Math. 143(2), 221–248 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Harcos, G.: Subconvex bounds for automorphic L-functions and applications. This is an unpublished dissertation available at http://www.renyi.hu/gharcos/ertekezes.pdf
  5. 5.
    Harcos, G.: An additive problem in the Fourier coefficients of cusp forms. Math. Ann. 326(2), 347–365 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence, RI (2004)Google Scholar
  7. 7.
    Iwaniec, H.: Spectral Methods of Automorphic Forms, 2 edn. Graduate Studies in Mathematics, vol. 53. American Mathematical Society, Providence, RI (2002)zbMATHGoogle Scholar
  8. 8.
    Kim, H.H.: Functoriality for the exterior square of \(GL_4\) and the symmetric fourth of \(GL_2\). J. Am. Math. Soc. 16(1), 139–183 (2003). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter SarnakCrossRefGoogle Scholar
  9. 9.
    Kowalski, E., Michel, P., VanderKam, J.: Rankin-Selberg \(L\)-functions in the level aspect. Duke Math. J. 114(1), 123–191 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lü, G.S.: On averages of Fourier coeffcients of Maass cusp forms. Arch. Math. 100, 255–265 (2013)CrossRefGoogle Scholar
  11. 11.
    Lü, G.S., Wu, J., Zhai, W.G.: Shifted convolution of cusp-forms with \(\theta \)-series. Ramanujan J. 40(1), C115–C133 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Luo, W.Z.: Shifted convolution of cusp-forms with \(\theta \)-series. Abh. Math. Semin. Univ. Ham- bg. 81, 45–53 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1944)zbMATHGoogle Scholar
  14. 14.
    Zhao, L.L.: The sum of divisors of a quadratic form. Acta Arith. 163(2), 161–177 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShandong UniversityJinanChina
  2. 2.School of Mathematics and StatisticsShandong UniversityWeihaiChina

Personalised recommendations