Advertisement

Ramanujan and dirichlet series with euler products

  • S. S. Rangachari
Article

Abstract

Ramanujan, in his unpublished manuscripts, had written down without proof, explicit linear combinations of cusp forms whose Mellin transforms possess Euler products in the sense of Hecke. All these results are proved here and their connection with the work of Hecke, Rankin and Serre is pointed out.

Keywords

Dirichlet scries Euler products Ramanujan’s work 

References

  1. [1]
    Birch B J 1975 A look back at Ramanujan’s Notebooks;Math. Proc. Cambridge Philos. Soc. 78, Part I pp. 73–79MATHMathSciNetCrossRefGoogle Scholar
  2. [2]a
    Hecke EMathematische Werke, Vandenhoeck and Ruprecht, 1959.Google Scholar
  3. [2]b
    (23)Zur Theorie der elliptischen Modulfunktionen (Math. Ann. 97 (1926), p. 210–242.Google Scholar
  4. [2]c
    (35, 36)Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktent wicklung. I, II (Math. Ann. 114 (1937), pp. 1–28, pp. 316–351).Google Scholar
  5. [2]d
    (41)Analytische Arithmetik der positiven quadratischen Formen. (Kgl. Danske Videnskabernes Selskab. Mathematisk-fysiske Meddelelser, XVII, 12 (1940), 134 pages)].Google Scholar
  6. [3]
    Li W W 1975 New forms and functional equations;Math. Ann. 212 4 285–315MATHCrossRefMathSciNetGoogle Scholar
  7. [4]
    Mordell L J 1916–19 On Mr. Ramanujan’s empirical expansions of modular functions;Proc. Cambridge Philos. Soc. 19 (1916–19), pp. 117–124Google Scholar
  8. [5]
    Newman M 1956 A table of the coefficients of the powers of η (τ)Indagationes Math. 18 204–216Google Scholar
  9. [6]
    Ramanathan K G 1980 Ramanujan and the congruence properties of partitions:Proc. Indian Acad. Sci. (Math. Sci.) 89 133–157MATHMathSciNetGoogle Scholar
  10. [7]
    Ramanujan S 1927On certain arithmetical functions. Collected papers of Srinivasa Ramanujan, Cambridge University Press, 136–162Google Scholar
  11. [8]
    Ramanujan S Unpublished manuscripts.Google Scholar
  12. [9]
    Rankin R A 1967 Hecke operators on congruence subgroups of the modular group;Math. Ann. 168 40–58MATHCrossRefMathSciNetGoogle Scholar
  13. [10]
    Rankin R A 1977Ramanujan’s unpublished work on congruences, Modular functions of one variable V. (eds) J P Serre and D B Zagier Lecture Notes 601 (New York: Springer-Verlag) 3–13Google Scholar
  14. [11]
    Serre J PModular forms of weight one and Galois representations, Durham symposium on algebraic number fields (L functions and Galois properties) (ed) A Fröhlich, (London: Academic Preys) 1977Google Scholar
  15. [12]
    Serre J P and Stark H M 1977Modular forms of weight 1/2. Modular functions of one variable VI. (eds) J P Serre and D B Zagier Lecture Notes 627 (New York: Springer Verlag), pp. 27–67CrossRefGoogle Scholar
  16. [13]
    Weber HLehrbuch der Algebra, Vol. III (New York: Chelsea)Google Scholar

Copyright information

© Indian Academy of Sciences 1982

Authors and Affiliations

  • S. S. Rangachari
    • 1
  1. 1.School of MathematicsTata Institute of Fundamental ResearchBombayIndia

Personalised recommendations