Skip to main content
SpringerLink
Log in

Ramanujan's unpublished work on congruences

  • Conference paper
  • First Online:
Book cover Modular Functions of one Variable V

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 601))

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B.J. Birch, A look back at Ramanujan's notebooks, Math. Proc. Cambridge Philos. Soc. 78(1975), 73–79.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Deligne, Formes modulaires et représentations l-adiques. Séminaire Bourbaki, 355, February 1969.

    Google Scholar 

  3. G.H. Hardy, Ramanujan; twelve lectures on subjects suggested by his life and work (Cambridge, 1940).

    Google Scholar 

  4. G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson (editors), Collected papers of Srinivasa Ramanujan (Cambridge, 1927).

    Google Scholar 

  5. E. Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Arch. Math. Phys. (3) 13(1908), 305–312.

    MATH  Google Scholar 

  6. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen (Leipzig, 1909).

    Google Scholar 

  7. W.J. LeVeque, Reviews in number theory, Amer. Math. Soc., Providence, R.I., 1974, Vol.2, F35.

    Google Scholar 

  8. M. Newman, Congruences for the coefficients of modular forms and some new congruences for the partition function, Canad. J. Math. 9(1957), 549–552.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22(1916), 159–184; no 18 in [4].

    Google Scholar 

  10. S. Ramanujan, Some properties of p(n), the number of partitions of n, Proc. Cambridge Philos. Soc. 19(1919), 207–210; no 25 in [4].

    MATH  Google Scholar 

  11. S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. (2) 18(1920), xix–xx; no 28 in [4].

    MathSciNet  MATH  Google Scholar 

  12. S. Ramanujan, Congruence properties of partitions, Math. Z. 9(1921), 147–153; no 30 in [4].

    Article  MathSciNet  Google Scholar 

  13. R.A. Rankin, George Neville Watson (obituary notice), J. London Math. Soc. 41(1966), 551–565.

    Article  MathSciNet  MATH  Google Scholar 

  14. R.A. Rankin, The divisibility of divisor functions, Proc. Glasgow Math. Assoc. 5(1961), 35–40.

    Article  MathSciNet  MATH  Google Scholar 

  15. J.M. Rushforth, Congruence properties of the partition function and associated functions, Ph.D. thesis, University of Birmingham, 1950.

    Google Scholar 

  16. J.M. Rushforth, Congruence properties of the partition function and associated functions, Proc. Cambridge Philos. Soc. 48(1952), 402–413.

    Article  MathSciNet  MATH  Google Scholar 

  17. E.J. Scourfield, On the divisibility of σv(n), Acta Arith. 10(1964), 245–285.

    MathSciNet  MATH  Google Scholar 

  18. J.-P. Serre, Une interprétation des congruences relatives à la fonction τ de Ramanujan. Séminaire Delange-Pisot-Poitou: 1967/68, Théorie des Nombres, Fasc. 1, Exp.14, 17 pp.

    Google Scholar 

  19. J.-P. Serre, Divisibilité des coefficients des formes modulaires de poids entier, C.R. Acad. Sci. Paris Sér.A 279(1974), 679–682.

    MATH  Google Scholar 

  20. J.-P. Serre, Divisibilité de certaines fonctions arithmétiques. Séminaire Delange-Pisot-Poitou: 1974/75, Théorie des Nombres, Exp.20, 28pp.

    Google Scholar 

  21. H.P.F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III, Lecture Notes in Mathematics, 350(1973), 1–55.

    Article  MathSciNet  Google Scholar 

  22. G.N. Watson, Über Ramanujansche Kongruenzeigenschaften der Zerfallungsanzahlen I, Math.Z. 39(1935), 712–731.

    Article  MathSciNet  Google Scholar 

  23. G.N. Watson, A table of Ramanujan's function τ(n), Proc. London Math. Soc. (2) 51(1948), 1–13.

    Google Scholar 

  24. J.R. Wilton, On Ramanujan's arithmetical function Σr,s(n), Proc. Cambridge Philos. Soc. 25(1929), 255–264.

    Article  MATH  Google Scholar 

  25. J.R. Wilton, Congruence properties of Ramanujan's function τ(n), Proc. London Math. Soc. (2) 31(1929), 1–10.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Glasgow, Glasgow

    R. A. Rankin

Authors
  1. R. A. Rankin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Jean-Pierre Serre Don Bernard Zagier

Rights and permissions

Reprints and permissions

Copyright information

© 1977 Springer-Verlag

About this paper

Cite this paper

Rankin, R.A. (1977). Ramanujan's unpublished work on congruences. In: Serre, JP., Zagier, D.B. (eds) Modular Functions of one Variable V. Lecture Notes in Mathematics, vol 601. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0063942

Download citation

  • DOI: https://doi.org/10.1007/BFb0063942

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08348-1

  • Online ISBN: 978-3-540-37291-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation