Skip to main content
SpringerLink
Log in

Higher power moments of shifted convolutions of Fourier coefficients involving squarefull kernel functions

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let f be a primitive holomorphic cusp form of even integral weight for the full modular group \(\Gamma =SL(2,\mathbb {Z})\). And let \(\lambda _{f}(n)\) be the nth normalized Fourier coefficient of f. Let a(n) be the function with squarefull kernel. Let \(j\ge 2\) be any fixed positive integer. In this paper, we establish the following result:

$$\begin{aligned} \sum _{n\le x}a(n)\lambda _{f}(n+1)^{2j} = C_{f,j}xP_{A_{j}-1}(\log x) + O\Big (x^{1-\frac{1}{2(3\cdot 2^{2j}-A_{j})}+\varepsilon }\Big ), \end{aligned}$$

where \(P_{A_{j}-1}(t)\) is polynomial of t with degree \(A_{j}-1\), and \(C_{f,j}>0\) is some suitable constant that can be explicitly evaluated.

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.: Shifted convolution sums and subconvexity bounds for automorphic \(L\)-functions. Int. Math. Res. Not. 73, 3905–3926 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. J. Am. Math. Soc. 30, 205–224 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Clozel, L., Thorne, J.A.: Level-raising and symmetric power functoriality. I. Compos. Math. 150, 729–748 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Clozel, L., Thorne, J.A.: Level-raising and symmetric power functoriality. II. Ann. Math. 181, 303–359 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Clozel, L., Thorne, J.A.: Level-raising and symmetric power functoriality. III. Duke Math. J. 166, 325–402 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deligne, P.: La Conjecture de Weil. I. Inst. Hautes Études Sci. Pull. Math. 43, 273–307 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erdös, P., Ivić, A.: The distribution of values of a certain class of arithmetic functions at consecutive integers. In: Number Theory, Budapest (Hungar), Colloq. Math. Soc. János Bolyai, vol. 51, pp. 45–91. North-Holland, Amsterdam (1987)

  8. Gelbart, S., Jacquet, H.: A relation between automorphic representations of \(GL(2)\) and \(GL(3)\). Ann. Sci. École Norm. Sup. 11, 471–542 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. Heath-Brown, D.R.: Hybrid bounds for \(L\)-functions. Invent. Math. 47, 149–170 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Holowinsky, R., Munshi, R.: Level aspect subconvexity for Rankin-Selberg L-functions. In: Automorphic Representations and L-functions, Stud. Math. Tata Inst. Fundam. Res., vol. 22, pp. 311–334 (2003)

  11. Ivić, A., Tenenbaum, G.: Local densities over integers free large prime factors. Quart. J. Math. Oxford. II Ser. 37, 401–417 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Iwaniec, H., Kowalski, E.: Analytic Number Theory, Amer. Math. Soc. Colloquium Publ., vol. 53. Amer. Math. Soc., Providence (2004)

  13. Kim, H., Shahidi, F.: Cuspidality of symmetric power with applications. Duke Math. J. 112, 177–197 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, H., Shahidi, F.: Functorial products for \(GL_{2}\times GL_{3}\) and functorial symmetric cube for \(GL_{2}\), with an appendix by C. J. Bushnell and G. Heniart. Ann. Math. 155, 837–893 (2002)

    Article  MathSciNet  Google Scholar 

  15. Kim, H.: Functoriality for the exterior square of \(GL_{4}\) and symmetric fourth of \(GL_{2}\), Appendix 1 by D. Ramakrishan, Appendix 2 by H. Kim and P. Sarnak. J. Am. Math. Soc. 16, 139–183 (2003)

    Article  Google Scholar 

  16. Lau, Y.-K., Lü, G.S.: Sums of Fourier coefficients of cusps forms. Q. J. Math. 62(3), 687–716 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lü, G.S., Wang, D.: Shifted convolution sums of Fourier coefficients with squarefull kernel functions. Lith. Math. J. 60, 51–63 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Michel, P.: The subconvexity probem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points. Ann. Math. 160(2), 185–236 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Newton, J., Thorne, J.A.: Symmetric power functoriality for holomorphic modular forms. Publ. Math. Inst. Hautes Études Sci. 134, 1–116 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Newton, J., Thorne, J.A.: Symmetric power functoriality for holomorphic modular forms. II. Publ. Math. Inst. Hautes Études Sci. 134, 117–152 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Perelli, P.: General \(L\)-functions. Ann. Math. Pura Appl. 130, 287–306 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shahidi, F.: Third symmetric power \(L\)-functions for \(GL(2)\). Compos. Math. 70, 245–273 (1989)

    MathSciNet  MATH  Google Scholar 

  23. Shahidi, F.: On certain \(L\)-functions. Am. J. Math. 103(2), 297–355 (1981)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to express his sincere gratitude to Professor Guangshi Lü and Professor Bin Chen for their constant encouragement and valuable suggestions. The author is extremely grateful to the anonymous referees for their meticulous checking, for thoroughly reporting countless typos and inaccuracies as well as for their valuable comments. These corrections and additions have made the manuscript clearer and more readable.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Weinan Normal University, Weinan, 714099, Shaanxi, China

    Guodong Hua

  2. School of Mathematics, Shandong University, Jinan, 250100, Shandong, China

    Guodong Hua

Authors
  1. Guodong Hua
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guodong Hua.

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 The National Key Research and Development Program of China (Grant No. 2021YFA1000700).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hua, G. Higher power moments of shifted convolutions of Fourier coefficients involving squarefull kernel functions. Ramanujan J 60, 585–596 (2023). https://doi.org/10.1007/s11139-022-00603-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-022-00603-2

Keywords

Mathematics Subject Classification

Search

Navigation