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Proceedings Mathematical Sciences

, Volume 104, Issue 1, pp 93–98 | Cite as

On Zagier’s cusp form and the Ramanujan τ function

  • Ashwaq Hashim
  • M. Ram Murty
Obituary note

Abstract

Zagier constructed a cusp form for each weightk of the full modular group. We use this construction to estimate Fourier coefficients of cusp forms. In particular, we get a non-trivial estimate, by elementary methods and indicate a relationship with the Lindelof hypothesis for classical Dirichlet L-functions.

Keywords

Ramanujan tau function cusp forms 

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References

  1. [B]
    Burgess D, On character sums andL-series II,Proc. Land. Math. Soc. (3) 13 (1963) 524–536MATHCrossRefMathSciNetGoogle Scholar
  2. [D]
    Deligne P, La conjecture de Weil I,Publ. I.H.E.S. 43 (1974) 273–307MathSciNetGoogle Scholar
  3. [GJ]
    Gelbart S and Jacquet H, A relation between automorphic representations ofGL(2) andGL(3),Ann. Sci. E. Norm. Sup. Ser. 11 (1978) 471–552MATHMathSciNetGoogle Scholar
  4. [H]
    Hecke E, Mathematische Werke, Herausgegeben im Auftrage der Akademie der Wissenschaften zu Göttingen, Vandenhoeck and Ruprecht, Göttingen, (1959) pp. 955Google Scholar
  5. [L]
    Langlands R, Problems in the theory of automorphic forms, in Lectures in Modern Analysis,Springer Lecture Notes 170 (1970) 18–26MathSciNetGoogle Scholar
  6. [GR]
    Gradshteyn I S and Ryzhik I M, Tables of integrals, series and products, 4th edition, Academic Press, (1965)Google Scholar
  7. [HB]
    Heath-Brown R, Hybrid bounds for DirichletL-functions II,Q. J. Math. Oxford (2)31 (1980) 157–167MATHCrossRefMathSciNetGoogle Scholar
  8. [Mu]
    Ram Murty M, On the oscillation of Fourier coefficients of modular forms.Math. Ann. 262 (1983) 431–446MATHCrossRefMathSciNetGoogle Scholar
  9. [Ra]
    Rademacher H, On the Phragmén-Lindelöf theorem and some applications,Math. Z. 72 (1959) 192–204MATHCrossRefMathSciNetGoogle Scholar
  10. [Se]
    Selberg A, On the estimation of Fourier coefficients of modular forms,Proc. Symp. Pure. Math. 8 (1965) 1–15MathSciNetGoogle Scholar
  11. [S]
    Shimura G, On the holomorphy of certain Dirichlet series,Proc. Lond. Math. Soc,31 (1975) 79–98MATHCrossRefMathSciNetGoogle Scholar
  12. [Sh]
    Shahidi F, On certainL-functions.Am. J. Math.,103 (1981) 297–355MATHCrossRefMathSciNetGoogle Scholar
  13. [Z]
    Zagier D, Modular forms whose Fourier coefficients involve zeta functions of quadratic fields, inModular functions of one variable VI, (eds. Serre J P and Zagier D B) Lecture Notes in Math., Vol. 627, pp. 105–170, Springer, 1977Google Scholar

Copyright information

© Indian Academy of Science 1994

Authors and Affiliations

  • Ashwaq Hashim
    • 1
  • M. Ram Murty
    • 1
  1. 1.Department of MathematicsMcGill UniversityMontrealCanada

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