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The Ramanujan Journal

, Volume 48, Issue 1, pp 1–11 | Cite as

Fourier coefficients of powers of the Dedekind eta function

  • Bernhard HeimEmail author
  • Markus Neuhauser
  • Florian Rupp
Article

Abstract

In this paper, we study the vanishing properties of Fourier coefficients of rth powers of the Dedekind eta function. We give a certain type of classification of this property based on r. Further we extend the results of Atkin, Cohen, and Newman for odd powers. This leads to a partial answer to a question raised by van Lint. We also indicate possible generalization of the Lehmer conjecture.

Keywords

Fourier coefficients Euler products Dedekind eta function Lehmer conjecture 

Mathematics Subject Classification

Primary 05A17 11F20 Secondary 11F30 11F37 

References

  1. 1.
    Bruinier, J.: Borchers Products on (2,l) and Chern Classes of Heegner Divisors. Lecture Notes in Mathematics, vol. 1780. Springer, New York (2002)Google Scholar
  2. 2.
    Heim, B.: A new type of functional equations of Euler products. In: Hagen, T., Rupp, F., Scheurle, J. (eds.) Dynamical Systems, Number Theory and Applications. A Festschrift in Honor of Armin Leutbecher’s 80th Birthday, Chapter 6, pp. 113–126. World Scientific Publishing, River Edge (2016)Google Scholar
  3. 3.
    Heim, B., Murase, A.: A characterization of Holomorphic Borcherds lifts by symmetries. Int. Math. Res. Not. (2014). doi: 10.1093/imm/rnv021
  4. 4.
    Koecher, M., Krieg, A.: Elliptische Funktionen und Modulformen. Springer, Berlin (2007)zbMATHGoogle Scholar
  5. 5.
    Köhler, G.: Eta Products and Theta Series Identities. Springer Monographs in Mathematics. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Lehmer, D.: The vanishing of Ramanujan’s \(\tau (n)\). Duke Math. J. 14, 429–433 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Miyake, T.: Modular Forms. Reprint of the 1989 English edn. Springer Monographs in Mathematics. Springer, Berlin (2006)Google Scholar
  8. 8.
    Newman, M.: An identity for the coefficients of certain modular forms. J. Lond. Math. Soc. 30, 488–493 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Newman, M.: A table of the coefficients of the powers of \(\eta (\tau )\). Proc. Acad. Amst. 59, 204–216 (1956)MathSciNetGoogle Scholar
  10. 10.
    Neher, E.: Jacobis Tripleprodukt-Identität und \(\eta \)-Identitäten in der Theorie der affinen Lie-Algebren. Jahresbericht d. Dt. Math.-Ver. 87, 164–181 (1985)Google Scholar
  11. 11.
    Ono, K.: A note on the Shimura correspondence and the Ramanujan \(\tau (n)\) function. Utilitas Math. 47, 170–180 (1995)MathSciNetGoogle Scholar
  12. 12.
    Ono, K., Robins, S.: Superlacunary Cusp forms. Proc. AMS 123(4), 1021–1029 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ono, K.: Gordon’s \(\epsilon \)-conjecture on the lacunarity of modular forms. C. R. Math. Acad. Sci. Soc. R. Can. 20, 103–107 (1998)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, vol. 102. Conference Board of Mathematical Sciences, Washington (2003)CrossRefGoogle Scholar
  15. 15.
    Ribet, K.: Galois Representations attached to Eigenforms with Nebentypus. In: Lecture Notes in Mathematics, vol. 601, pp. 17–52 (1977)Google Scholar
  16. 16.
    Serre, J.: Sur la lacunarité des puissances de \(\eta \). Glasgow Math. J. 27, 203–221 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shimura, G.: Introduction to the Arithmetical Theory of Automorphic Functions. Iwanami Shoten and Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  18. 18.
    Shimura, G.: On modular forms of half-integral weight. Ann. Math. 97, 440–481 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    van Lint, J.H.: Hecke operators and Euler products. Thesis, Utrecht (1957)Google Scholar
  20. 20.
    Wohlfahrt, K.: Über Operatoren Heckescher Art bei Modulformen reeller Dimension. Math. Nachr. 16, 233–256 (1957)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.German University of Technology in OmanMuscatSultanate of Oman

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