Skip to main content
SpringerLink
Log in

An exponential sum involving Fourier coefficients of eigenforms for \(SL(2,\pmb {{\mathbb {Z}}})\)

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let \(\lambda _f (n)\) denote the normalized n-th Fourier coefficient of a holomorphic Hecke eigencuspform or a Hecke–Maass cusp form for the full modular group. In this paper we shall exhibit cancellations in the following sum:

$$\begin{aligned} \sum _{N<n \leqslant 2N} \lambda _f(n) \nu (n) e\left( \alpha n^{\theta }\right) , \end{aligned}$$

where \(\alpha , \ \theta \) are real numbers with \(0< \theta <1\), and \(\nu (n)\) is either \(\mu (n)\) or \(\Lambda (n)\).

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. Adhikari, S.D.: \(\Omega \)-Results for sums of Fourier coefficients of cusp forms. Acta. Arith. 57, 83–92 (1991)

    Article  MathSciNet  Google Scholar 

  2. Deligne, P.: La conjecture de Weil. I. Inst. Ht. Etudes Sci. Publ. Math. 43, 273–307 (1974)

    Article  MathSciNet  Google Scholar 

  3. Dunster, T.M.: Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21, 995–1018 (1990)

    Article  MathSciNet  Google Scholar 

  4. Fouvry, E., Ganguly, S.: Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms. Compos. Math. 150, 763–797 (2014)

    Article  MathSciNet  Google Scholar 

  5. Friedlander, J.B., Iwaniec, H.: Summation formulae for the coefficients of \(L-\) functions. Can. J. Math. 57(3), 494–505 (2005)

    Article  MathSciNet  Google Scholar 

  6. Godber, D.: Additive twists of Fourier coefficients of modular forms. J. Number Theory 133, 83–104 (2013)

    Article  MathSciNet  Google Scholar 

  7. Harcos, G., Michel, P.: The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II. Invent. Math 163(3), 581–655 (2006)

    Article  MathSciNet  Google Scholar 

  8. Hou, F.: On exponential sum involving Fourier coefficients of cusp forms over primes. J. Number Theory 164, 375–404 (2016)

    Article  MathSciNet  Google Scholar 

  9. Huxley, M.N.: Introduction to Kloostermania, in Elementary and Analytic Theory of Numbers, vol. 17, pp. 217–306. Banach Center Publication, Warsaw (1985)

    Google Scholar 

  10. Ivić, A.: The Riemann Zeta-Function: Theory and Applications. Dover Publications Inc., New York (2012)

    MATH  Google Scholar 

  11. Iwaniec, H.: Topics in Classical Automorphic Forms, Graduate Text in Mathematics, vol. 17. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  12. Iwaniec, H.: Spectral Methods of Automorphic Forms. Graduate Studies in Mathematics. Revista Matemàtica Iberoamericana, vol. 53. American Mathematical Society, Providence (2002)

    MATH  Google Scholar 

  13. Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publication, vol. 53. American Mathematical Society, Providence (2004)

    MATH  Google Scholar 

  14. Iwaniec, H., Luo, W., Sarnak, P.: Low lying zeros of family of \(L\)-functions. Inst. Ht. Études Sci. Publ. Math. 91, 55–131 (2000)

    Article  Google Scholar 

  15. Kim, H., Sarnak, P.: Refined estimates towards the Ramanujan and Selberg conjectures. J. Am. Math. Soc. 16, 175–181 (2003)

    Google Scholar 

  16. Murty, M.R.: Oscillation of Fourier coefficients of modular forms. Math. Ann. 262, 431–446 (1983)

    Article  MathSciNet  Google Scholar 

  17. Pi, Q., Sun, Q.: Oscillations of cusp form coefficients over primes. Acta. Appl. Math. 109, 1155–1163 (2010)

    Article  MathSciNet  Google Scholar 

  18. Pi, Q., Sun, Q.: Oscillations of cusp form coefficients in exponential sums. Ramanujan J. 21, 53–64 (2010)

    Article  MathSciNet  Google Scholar 

  19. Zhao, L.: Oscillations of Hecke eigenvalues at shifted primes. Rev. Mat. Iberoam. 22, 323–337 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Satadal Ganguly and Ritabrata Munshi for helpful suggestions and comments. They also thank ISI Kolkata for wonderful academic atmosphere.

Author information

Authors and Affiliations

  1. Stat-Math Unit, Indian Statistical Institute, 203 BT Road, Calcutta, 700108, India

    Ratnadeep Acharya & Saurabh Kumar Singh

Authors
  1. Ratnadeep Acharya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saurabh Kumar Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ratnadeep Acharya.

Additional information

Publisher's Note

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

During the work, S. Singh was supported by the Department of Atomic Energy, Government of India, NBHM post doctoral fellowship No: 2/40(15)/2016/R&D-II/5765.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Acharya, R., Singh, S.K. An exponential sum involving Fourier coefficients of eigenforms for \(SL(2,\pmb {{\mathbb {Z}}})\). Ramanujan J 54, 699–716 (2021). https://doi.org/10.1007/s11139-020-00255-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-020-00255-0

Keywords

Mathematics Subject Classification

Search

Navigation