SpringerLink
Log in

Integral mean-square estimation of error terms of Fourier coefficients attached to cusp forms

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

Let f, g be two distinct primitive holomorphic cusp forms of even integral weights \(k_{1},k_{2}\) for the full modular group \(\Gamma =SL(2,\mathbb {Z})\), respectively. In this paper, we are interested in the upper bounds for the integral mean square of error terms involving \(\lambda _{f}^{2i_{1}}(n), \lambda _{f}^{2}(n^{j_{1}})\) and \(\lambda _{f}^{2}(n^{i_{2}})\lambda _{g}^{2}(n^{j_{2}})\) on average in terms of f, g respectively. Here \(i_{1}\ge 2, j_{1}\ge 3\) and \(i_{2},j_{2}\ge 1\) are positive integers.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Thailand)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aggarwal K, A new subconvex bound for \(GL(3)\)\(L\)-functions in the \(t\)-aspect, Int. J. Number Theory 17 (2021) 1111–1138

    Article  MathSciNet  MATH  Google Scholar 

  2. Barnet-Lamb T, Geraghty D, Harris M and Taylor R, A family of Calabi–Yau varieties and potential automorphy. II, Publ. Res. Inst. Math. Sci. 47(1) (2011) 29–98

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

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

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

  8. Fomenko O M, Fourier coefficients of parabolic forms and automorphic \(L\)-functions, J. Math. Sci. (New York) 95(3) (1999) 2295–2316

    Article  MathSciNet  Google Scholar 

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

  10. Hecke E, Theorie der Eisensteinsche Reihen höherer Stufe und ihre Anwendung auf Funktionentheorie und Arithmetik, Abh. Math. Sem. Univ. Hamburg 5(1) (1927) 199–224

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang B R, On the Rankin–Selberg problem, Math. Ann. 381(3/4) (2021) 1217–1251

    Article  MathSciNet  MATH  Google Scholar 

  12. Ivić A, Matsumoto K and Tanigawa Y, On Riesz means of the coefficients of the Rankin–Selberg series, Math. Proc. Cambridge Philos. Soc. 127 (1999) 117–131

    Article  MathSciNet  MATH  Google Scholar 

  13. Ivić A, On some mean square estimates in the Rankin–Selberg problem, Appl. Anal. Discrete Math. 1 (2007) 111–121

    Article  MathSciNet  MATH  Google Scholar 

  14. Ivić A, The Riemann zeta-function, Theory and applications, Reprint of the 1985 original (2003) New York: Wiley; Mineola, New York: Dover Publications Inc.) xxii+517

  15. Iwaniec H and Kowalski E, Analytic Number Theory, Amer. Math. Soc. Colloquium Publ., Vol. 53, Amer. Math. Soc., Providence, RI (2004)

  16. Jacquet H and Shalika J A, On the Euler products and the classification of automorphic representations I, Amer. J. Math. 103 (1981) 499–558

    Article  MathSciNet  MATH  Google Scholar 

  17. Jacquet H and Shalika J A, On the Euler products and the classification of automorphic forms II, Amer. J. Math. 103 (1981) 777–815

    Article  MathSciNet  MATH  Google Scholar 

  18. Jacquet H, Piatetski-Shapiro I I and Shalika J A, Rankin–Selberg convolutions. I, Amer. J. Math. 105(2) (1983) 367–464

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Kim H and 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 (2002) 837–893

    Article  MathSciNet  Google Scholar 

  21. 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. Amer. Math. Soc. 16 (2003) 139–183

    Article  MathSciNet  Google Scholar 

  22. Lao H X and Sankaranarayanan A, On the mean-square of the error term related to \(\sum _{n\le x}\lambda ^{2}(n^{j})\), Sci. China Math. 54 (2011) 855–864

    Article  MathSciNet  MATH  Google Scholar 

  23. Lao H X and Sankaranarayanan A, Integral mean square estimation for the error term related to \(\sum _{n\le x}\lambda ^{2}(n^{2})\), Sci. China Math. 58(10) (2015) 2125–2132

    Article  MathSciNet  MATH  Google Scholar 

  24. Lao H X and Luo S, Sign changes and non-vanishing of Fourier coefficients of holomorphic cusp forms, Rocky Mountain J. Math. 51(5) (2021) 1701–1714

    Article  MathSciNet  MATH  Google Scholar 

  25. Lau Y-K and Lü G S, Sums of Fourier coefficients of cusp forms, Quart. J. Math. 62 (2011) 687–716

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu J Y and Ye Y B, Perron’s formula and the prime number theorem for automorphic \(L\)-functions, in: Pure Appl. Math. Q., Special Issue: In honor of Leon Simon, Part 1, 3 (2007) No. 2, pp. 481–497.

  27. Luo S, Lao H X and Zou A Y, Asymptotics for Dirichlet coefficients of symmetric power \(L\)-functions, Acta Math. 199 (2021) 253–268

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Ramachandra K and Sankaranarayanan A, Notes on the Riemann-zeta function, J. Indian Math. Soc. (N.S.) 57 (1991) 67–77

  32. Rankin R A, Contributions to the theory of Ramanujan’s function \(\tau (n)\) and similar arithmetical functions II, The order of the Fourier coefficients of the integral modular forms, Proc. Cambridge Philos. Soc. 35 (1939) 357–372

    Article  MATH  Google Scholar 

  33. Rankin R A, Sums of cusp form coefficients, in: Automorphic forms and analytic number theory (Montreal, PQ, 1989) (1990) (Montreal, QC: Univ. Montreal) pp. 115–121

  34. Runick Z and Sarnak P, Zeros of principle \(L\)-functions and ranom matrix theory, Duke Math. J. 81(2) (1996) 269–322

    MathSciNet  Google Scholar 

  35. Selberg A, Bemerkungen über eine Dirichletsche, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940) 47–50

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Shahidi F, Fourier transform of intertwining operators and Plancherel measures for \(GL(n)\), Amer. J. Math. 106(1) (1984) 67–111

    Article  MathSciNet  MATH  Google Scholar 

  38. Shahidi F, Local coefficients as Artin factors for real groups, Duke Math. J. 52(4) (1985) 973–1007

    Article  MathSciNet  MATH  Google Scholar 

  39. Shahidi F, A proof of Langland’s conjecture on Plancherel measures; complementary series for \(p\)-adic groups, Ann. Math. 132(2) (1990) 273–330

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  41. Wilton J R, A note on Ramanujan’s arithmetical function \(\tau (n)\), Proc. Cambridge Philos. Soc. 25 (1928) 121–129

    Article  MATH  Google Scholar 

  42. Wu J, Power sums of Hecke eigenvalues and application, Acta Arith. 137(4) (2009) 333–344

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to express his gratitude to Professors Guangshi Lü, Bin Chen, Bingrong Huang and Research Fellow, Zhiwei Wang 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 readable. This work was financially supported in part by The National Key Research and Development Program of China (Grant No. 2021YFA1000700), Natural Science Basic Research Program of Shaanxi (Program Nos 2023-JC-QN-0024 and 2023-JC-YB-077), Foundation of Shaanxi Educational Committee (2023-JCYB- 013) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSQ010).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Chaoyang Campus, Weinan Normal University, Weinan, 714 099, China

    Guodong Hua

  2. School of Mathematics, Central Campus, Shandong University, **an, 250 100, 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

Communicating Editor: Sanoli Gun

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hua, G. Integral mean-square estimation of error terms of Fourier coefficients attached to cusp forms. Proc Math Sci 133, 17 (2023). https://doi.org/10.1007/s12044-023-00737-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12044-023-00737-3

Keywords

2010 Mathematics Subject Classification

Search

Navigation