Mathematische Zeitschrift

, Volume 23, Issue 1, pp 1–37 | Cite as

Some problems of ‘Partitio numerorum’ (VI): Further researches in Waring's Problem

  • G. H. Hardy
  • J. E. Littlewood
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. 2).
    That is to say, there is a constantA such that more thanAn numbers less thann are sums of 7 cubes. This result is due to Baer (W. S. Baer, Über die Zerlegung der ganzen Zahlen in sieben Kuben, Mathematische Annalen74 (1913), 511–514).Google Scholar
  2. 3).
    E. Landau, Über die Anzahl der Gitterpunkte in gewissen Bereichen, Göttinger Nachrichten (1912), 687–771 (750).Google Scholar
  3. 4).
    A. Hurwitz, Über die Darstellung der ganzen Zahlen als Summen vonn-ten Potenzen ganzer Zahlen, Mathematische Annalen65 (1908), 424–427.Google Scholar
  4. 5).
    See P. N. 4, 179 (Theorem 4). The ground of the inequalityG (k)≧Γ(k) being (except fork=4) the existence of a forbidden arithmetical progression, the same inequality holds forG 1 (k).Google Scholar
  5. 9).
    See P. N. 4, 179, f. n. 28) Compare P. N. 5, 52 (Lemma 10). This point aboutk=4 is overlooked on p. 188 (last sentence).Google Scholar
  6. 10).
    H. Weyl, Bemerkung über die Hardy-Littlewoodschen Untersuchungen zum Waringschen Problem, Göttinger Nachrichten 1921, 189–192.Google Scholar
  7. 11).
    E. Landau, Zum Waringschen Problem, Hilbert Festschrift (1922), 423–451 [Math. Zeitschr.12 (1921), 219–247].Google Scholar
  8. 11a).
    This notation means that we sum forq≦ν and for all values ofp associated with each suchq.Google Scholar
  9. 13).
    Our mistake there lay in failing to distinguish the two cases.Google Scholar
  10. 14).
    We owe this observation to Prof. Landau.Google Scholar
  11. 15).
    P. N. 4, 176, (4. 14) and the equation five lines lower.Google Scholar
  12. 16).
    P. N. 4, 177, (4. 22).Google Scholar
  13. 17).
    Compare P. N. 5, 49 (Lemma 5). The argument there is simpler.Google Scholar
  14. 18).
    P. N. 2, 18.Google Scholar
  15. 19).
    P. N. 2, 19–21.Google Scholar
  16. 20).
    Here, and in the formulae which follow, the values of thep's, which differ from formula to formula, are irrelevant.Google Scholar
  17. 21).
    P. N. 4, 175.Google Scholar
  18. 22).
    P. N. 4, 166, Lemma 1.Google Scholar
  19. 23).
    Compare P. N. 5, 52, Lemma 6.Google Scholar
  20. 24).
    Thus, ifk=15 andq=22·3·52·72·11,q 1=11,q 2=72,Q=22·3·52.Google Scholar
  21. 25).
    P. N. 2, 20; P. N. 4, 170.Google Scholar
  22. 26).
    We use for the moment the notation of P. N. 1, except naturally that, in conformity with the conventions laid down in § 1.1, we writeB instead ofA.Google Scholar
  23. 27).
    Landau, l. c. 226 (Hilfssatz 2). Zum Waringschen Problem, Hilbert Festschrift (1922), 423–451 [Math. Zeitschr.12 (1921), 219–247].Google Scholar
  24. 27).
    β is ultimately chosen in three different ways to suit three different arguments. In each case it is chosen as a function ofk ands only. We anticipate this choice and allow ourselves to treat β as ac.Google Scholar
  25. 28).
    Compare P. N. 5, 52 (Lemma 10).Google Scholar
  26. 29).
    Compare P. N. 5, 52 (Lemma 11).Google Scholar
  27. 30).
    The dash denoting that any term for whichmQ+h=0 is to be omitted.Google Scholar
  28. 31).
    Landau, l. c 230 (Hilfssatz 4) Zum Waringschen Problem, Hilbert Festschrit (1922), 423–451 [Math. Zeitschr.12 (1921), 219–247].Google Scholar
  29. 32).
    See P. N. 2, 16–17; Landau, l. c. 241. Zum Waringschen Problem, Hilbert Festschrift (1922) 423–451 [Math. Zeitschr12 (1921), 219–247].Google Scholar
  30. 33).
    P. N. 4, 179.Google Scholar
  31. 34).
    We usea momentarily for an indicialA (i. e. an absolutec). We may plainly supposea<1.Google Scholar
  32. 35).
    It may be shown that no other choice ofs ands′ leads to a better value ofs+s′.Google Scholar
  33. 35a).
    This is the special case of (6.31) in whicht=0.Google Scholar
  34. 36).
    Valid since 15>8+2k.Google Scholar
  35. 38).
    See P. N. 4, 184, for an analysis of the exceptional cases.Google Scholar
  36. 39).
    In fact, in the notation of § 7,a s=1 (sk).Google Scholar

Copyright information

© Springer-Verlag 1925

Authors and Affiliations

  • G. H. Hardy
    • 1
  • J. E. Littlewood
    • 2
  1. 1.Oxford
  2. 2.Cambridge

Personalised recommendations