Skip to main content
SpringerLink
Log in

Triple correlations of Fourier coefficients of cusp forms

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms \(\sum _{H\le h\le 2H}W\left( \frac{h}{H}\right) \sum _{X\le n\le 2X}\lambda _{1}(n-h)\lambda _{2}(n)\lambda _{3}(n+h)\), which is nontrivial provided that \(H\ge X^{2/3+\varepsilon }\). The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.

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. Blomer, V.: On triple correlations of divisor functions. Bull. Lond. Math. Soc. (to appear)

  2. Blomer, V., Milićević, D.: The second moment of twisted modular L-functions. Geom. Funct. Anal. 25(2), 453–516 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blomer, V., Harcos, G., Michel, P.: A Burgess-like subconvex bound for twisted L-functions. Forum Math. 19(1), 61–105 (2007). Appendix 2 by Z. Mao

    Article  MathSciNet  MATH  Google Scholar 

  4. Blomer, V., Fouvry, É., Kowalski, E., Michel, P., Milićević, D.: On moments of twisted L-functions. Am. J. Math. (to appear)

  5. Browning, T.: The divisor problem for binary cubic forms. Journal de Théorie des nombres de Bordeaux 23(3), 579–602 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70(2), 219–288 (1982/83)

  7. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic Press, Amsterdam (2007). Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX)

  8. Jutila, M.: A variant of the circle method, Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995). London Mathematical Society Lecture Note Series, vol. 237, pp. 245–254. Cambridge University Press, Cambridge (1997)

  9. Jutila, M.: Convolutions of Fourier coefficients of cusp forms. Publ. Inst. Math. (Beograd) (N.S.) 65(79), 31–51 (1999)

  10. Kowalski, E., Michel, P., VanderKam, J.: Rankin–Selberg L-functions in the level aspect. Duke Math. J. 114(1), 123–191 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Meurman, T.: On the Binary Additive Divisor Problem, Number Theory (Turku, 1999), de Gruyter. Berlin, pp. 223–246 (2001)

Download references

Acknowledgements

The author is grateful to Valentin Blomer for several helpful comments and suggestions during the preparation of this article. He also thanks Sheng-Chi Liu for drawing his attention to [1] and Roman Holowinsky for his encouragement and comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH, 43210-1174, USA

    Yongxiao Lin

Authors
  1. Yongxiao Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongxiao Lin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, Y. Triple correlations of Fourier coefficients of cusp forms. Ramanujan J 45, 841–858 (2018). https://doi.org/10.1007/s11139-016-9874-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-016-9874-1

Keywords

Mathematics Subject Classification

Search

Navigation