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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Ramanujan, in his unpublished manuscripts, had written down without proof, explicit linear combinations of cusp forms whose Mellin transforms possess Euler products in the sense of Hecke.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.

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References

  1. Birch B J 1975 A look back at Ramanujan’s Notebooks;Math. Proc. Cambridge Philos. Soc. 78, Part I pp. 73–79

    Article  MATH  MathSciNet  Google Scholar 

  2. Hecke EMathematische Werke, Vandenhoeck and Ruprecht, 1959.

  3. (23)Zur Theorie der elliptischen Modulfunktionen (Math. Ann. 97 (1926), p. 210–242.

  4. (35, 36)Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktent wicklung. I, II (Math. Ann. 114 (1937), pp. 1–28, pp. 316–351).

  5. (41)Analytische Arithmetik der positiven quadratischen Formen. (Kgl. Danske Videnskabernes Selskab. Mathematisk-fysiske Meddelelser, XVII, 12 (1940), 134 pages)].

  6. Li W W 1975 New forms and functional equations;Math. Ann. 212 4 285–315

    Article  MATH  MathSciNet  Google Scholar 

  7. Mordell L J 1916–19 On Mr. Ramanujan’s empirical expansions of modular functions;Proc. Cambridge Philos. Soc. 19 (1916–19), pp. 117–124

    Google Scholar 

  8. Newman M 1956 A table of the coefficients of the powers of η (τ)Indagationes Math. 18 204–216

    Google Scholar 

  9. Ramanathan K G 1980 Ramanujan and the congruence properties of partitions:Proc. Indian Acad. Sci. (Math. Sci.) 89 133–157

    MATH  MathSciNet  Google Scholar 

  10. Ramanujan S 1927On certain arithmetical functions. Collected papers of Srinivasa Ramanujan, Cambridge University Press, 136–162

  11. Ramanujan S Unpublished manuscripts.

  12. Rankin R A 1967 Hecke operators on congruence subgroups of the modular group;Math. Ann. 168 40–58

    Article  MATH  MathSciNet  Google Scholar 

  13. Rankin R A 1977Ramanujan’s unpublished work on congruences, Modular functions of one variable V. (eds) J P Serre and D B Zagier Lecture Notes 601 (New York: Springer-Verlag) 3–13

    Google Scholar 

  14. 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) 1977

    Google Scholar 

  15. 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. 27–67

    Chapter  Google Scholar 

  16. Weber HLehrbuch der Algebra, Vol. III (New York: Chelsea)

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Rangachari, S.S. Ramanujan and dirichlet series with euler products. Proc. Indian Acad. Sci. (Math. Sci.) 91, 1–15 (1982). https://doi.org/10.1007/BF02837257

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  • DOI: https://doi.org/10.1007/BF02837257

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