Skip to main content

Euler Products in Ramanujan’s Lost Notebook

  • Chapter
  • First Online:
Ramanujan's Lost Notebook

Abstract

In his famous paper, On certain arithmetical functions [229], Ramanujan offers for the first time the Euler product for what is now known as Ramanujan’s Dirichlet series.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part III, Springer, New York, 2012.

    Google Scholar 

  2. B.C. Berndt, B. Kim, and K.S. Williams, Euler products in Ramanujan’s lost notebook, Internat. J. Number Thy. 9 (2013), 1313–1349.

    Article  MathSciNet  Google Scholar 

  3. B.C. Berndt and K. Ono, Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary, Sém. Lotharingien de Combinatoire 42 (1999), 63 pp.; in The Andrews Festschrift, D. Foata and G.–N. Han, eds., Springer–Verlag, Berlin, 2001, pp. 39–110.

    Google Scholar 

  4. H.H. Chan, S. Cooper, and W.–C. Liaw, On η 3 (aτ)η 3 (bτ) with a + b = 8, J. Austral. Math. Soc. 84 (2008), 301–313.

    Google Scholar 

  5. H.H. Chan, S. Cooper, and P.C. Toh, Ramanujan’s Eisenstein series and powers of Dedekind’s eta-function, J. London Math. Soc. (2) 75 (2007), 225–242.

    Google Scholar 

  6. H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires in Modular Functions of One Variable VI, Lecture Notes in Math., No. 627, Springer-Verlag, Berlin, 1977, pp. 69–78.

    Google Scholar 

  7. P. Deligne, La conjecture de Weil. I., Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307.

    Google Scholar 

  8. F. Diamond and J. Shurman, A First Course in Modular Forms, Springer, New York, 2005.

    MATH  Google Scholar 

  9. E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, I, II, Math. Ann. 114 (1937), 1–28, 316–351.

    Article  MathSciNet  Google Scholar 

  10. E. Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1970.

    MATH  Google Scholar 

  11. L.K. Hua, Introduction to Number Theory, Springer-Verlag, Berlin, 1982.

    Google Scholar 

  12. J.G. Huard, P. Kaplan, and K.S. Williams, The Chowla–Selberg formula for genera, Acta Arith. 73 (1995), 271–301.

    Article  MathSciNet  Google Scholar 

  13. H. Iwaniec, Topics in Classical Automorphic Forms, American Mathematical Society, Providence, RI, 1997.

    Book  Google Scholar 

  14. F. Klein and R. Fricke, Vorlesungen über die Theorie der Elliptischen Modulfunctionen, Bd. 2, Teubner, Leipzig, 1892.

    Google Scholar 

  15. N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1984.

    Book  Google Scholar 

  16. G. Köhler, Eta Products and Theta Series Identities, Springer, Berlin, 2011.

    Book  Google Scholar 

  17. Y. Martin, Multiplicative η-products, Trans. Amer. Math. Soc. 348 (1996), 4825–4856.

    Article  MathSciNet  Google Scholar 

  18. L.J. Mordell, On Mr. Ramanujan’s empirical expansions of modular functions, Proc. Cambridge Philos. Soc. 19(1917), 117–124.

    MATH  Google Scholar 

  19. M. Newman, Modular forms whose coefficients possess multiplicative properties, Ann. Math. 70 (1959), 478–489.

    Article  MathSciNet  Google Scholar 

  20. M. Newman, Construction and application of a class of modular functions II, Proc. London Math. Soc. (3) 9 (1959), 373–387.

    Google Scholar 

  21. K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-series, American Mathematical Society, Providence, RI, 2004.

    MATH  Google Scholar 

  22. S. Raghavan, On Ramanujan and Dirichlet series with Euler products, Glasgow Math. J. 25 (1984), 203–206.

    Article  MathSciNet  Google Scholar 

  23. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159–184.

    Google Scholar 

  24. S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927; reprinted by Chelsea, New York, 1962; reprinted by the American Mathematical Society, Providence, RI, 2000.

    Google Scholar 

  25. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.

    Google Scholar 

  26. S.S. Rangachari, Ramanujan and Dirichlet series with Euler products, Proc. Indian Acad. Sci. (Math. Sci.) 91 (1982), 1–15.

    Article  MathSciNet  Google Scholar 

  27. S.S. Rangachari, Euler products, modular identities and elliptic integrals in Ramanujan’s manuscripts, II, in Ramanujan Revisited, G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan, and R.A. Rankin, eds., Academic Press, Boston, 1988, pp. 347–357.

    Google Scholar 

  28. J.-P. Serre, Modular forms of weight one and Galois representations, in Algebraic Number Fields: L-functions and Galois Properties (Proc. Sympos., Univ. Durham, 1975), Academic Press, London, 1977, pp. 193–268.

    Google Scholar 

  29. Z.-H. Sun, The expansion of ∏ k=1 (1 − q ak )(1 − q bk ), Acta Arith. 134 (2008), 11–29.

    Article  MathSciNet  Google Scholar 

  30. Z.-H. Sun and K.S. Williams, On the number of representations of n by ax 2 + bxy + cy 2, Acta Arith. 122 (2006), 101–171.

    Article  MathSciNet  Google Scholar 

  31. Z.-H. Sun and K.S. Williams, Ramanujan identities and Euler products for a type of Dirichlet series, Acta Arith. 122 (2006), 349–393.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Andrews, G.E., Berndt, B.C. (2018). Euler Products in Ramanujan’s Lost Notebook. In: Ramanujan's Lost Notebook. Springer, Cham. https://doi.org/10.1007/978-3-319-77834-1_15

Download citation

Publish with us

Policies and ethics