Skip to main content

Some problems of ‘Partitio Numerorum’: IV. The singular series in Waring’s Problem and the value of the number G (k)

  • Chapter
Book cover Festschrift

Zusammenfassung

In this memoir we continue the investigations initiated in two earlier memoirs bearing a similar title, and complete the proof of all the assertions which they contain1). We shall assume throughout that the reader is acquainted with the notation and terminology of these memoirs.

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 59.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 17.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 29.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio Numerorum’: I.A new solution of Waring’s Problem, Göttinger Nachrichten 1920, S. 33–54;

    Google Scholar 

  2. Proof that every large number is the sum of at most 21 biquadrates, Mathematische Zeitschrift 9 (1921), S. 14–27.

    Google Scholar 

  3. D. Hilbert, Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen, Göttinger Nachrichten 1909, S. 17–36

    Google Scholar 

  4. reprinted with certain changes in Mathematische Annalen, 67 (1909), S. 281–300.

    Google Scholar 

  5. E. Landau, Zur Hardy-Littlewoodschen Lösung des Waringschen Problems, Göttinger Nachrichten 1921, S. 88–92.

    Google Scholar 

  6. H. Weyl, Bemerkung zur Hardy-Littlewoodschen Lösung des Waringschen Problems, Göttinger Nachrichten 1922.

    Google Scholar 

  7. A. Ostrowski, Bemerkung zur Hardy-Littlewoodschen Lösung des Waringschen Problems, Mathematische Zeitschrift, 9 (1921), S. 28–34. We return to this point in § 6. 3.

    Article  MathSciNet  Google Scholar 

  8. E. Landua, Über eine Anwendung der Primzahltheorie auf das Waringsche Problem in der elementaren Zahlentheorie, Mathematische Annalen, 66 (1909), S. 102–105.

    Article  Google Scholar 

  9. A. Hurwitz, Über die Darstellung der ganzen Zahlen als Summen von n-ter Potenzen ganzer Zahlen, Mathematische Annalen, 65, (1908), S. 424–427.

    Article  MathSciNet  MATH  Google Scholar 

  10. E.Maillet, Sur la decomposition d’un entire en une somme de puissances huitièmes d’entiers, Bulletin de la société mathématique de France, 36 (1908), p. 69–77.

    MathSciNet  MATH  Google Scholar 

  11. See S. Ramanujan, On certain trigonometrical sums and their applications in the theory of numbers, Transactions of the Cambridge Philosophical Society, 22 (1918), pp. 259–276

    Google Scholar 

  12. G. H. Hardy, Note on Ramanujan’s function c q (n), Proceedings of the Cambridge Philosophical Society, 20 (1921), pp. 263–271; and P. N. 3.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1982 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Hardy, G.H., Littlewood, J.E. (1982). Some problems of ‘Partitio Numerorum’: IV. The singular series in Waring’s Problem and the value of the number G (k). In: Festschrift. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61810-9_39

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-61810-9_39

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-61811-6

  • Online ISBN: 978-3-642-61810-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics