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

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Literatur

  1. 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).

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  2. E. Landau, Über die Anzahl der Gitterpunkte in gewissen Bereichen, Göttinger Nachrichten (1912), 687–771 (750).

  3. A. Hurwitz, Über die Darstellung der ganzen Zahlen als Summen vonn-ten Potenzen ganzer Zahlen, Mathematische Annalen65 (1908), 424–427.

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  4. 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).

  5. 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).

  6. H. Weyl, Bemerkung über die Hardy-Littlewoodschen Untersuchungen zum Waringschen Problem, Göttinger Nachrichten 1921, 189–192.

  7. E. Landau, Zum Waringschen Problem, Hilbert Festschrift (1922), 423–451 [Math. Zeitschr.12 (1921), 219–247].

  8. This notation means that we sum forq≦ν and for all values ofp associated with each suchq.

  9. Our mistake there lay in failing to distinguish the two cases.

  10. We owe this observation to Prof. Landau.

  11. P. N. 4, 176, (4. 14) and the equation five lines lower.

  12. P. N. 4, 177, (4. 22).

  13. Compare P. N. 5, 49 (Lemma 5). The argument there is simpler.

  14. P. N. 2, 18.

  15. P. N. 2, 19–21.

  16. Here, and in the formulae which follow, the values of thep's, which differ from formula to formula, are irrelevant.

  17. P. N. 4, 175.

  18. P. N. 4, 166, Lemma 1.

  19. Compare P. N. 5, 52, Lemma 6.

  20. Thus, ifk=15 andq=22·3·52·72·11,q 1=11,q 2=72,Q=22·3·52.

  21. P. N. 2, 20; P. N. 4, 170.

  22. 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.

  23. Landau, l. c. 226 (Hilfssatz 2). Zum Waringschen Problem, Hilbert Festschrift (1922), 423–451 [Math. Zeitschr.12 (1921), 219–247].

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  24. 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.

  25. Compare P. N. 5, 52 (Lemma 10).

  26. Compare P. N. 5, 52 (Lemma 11).

  27. The dash denoting that any term for whichmQ+h=0 is to be omitted.

  28. Landau, l. c 230 (Hilfssatz 4) Zum Waringschen Problem, Hilbert Festschrit (1922), 423–451 [Math. Zeitschr.12 (1921), 219–247].

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  29. See P. N. 2, 16–17; Landau, l. c. 241. Zum Waringschen Problem, Hilbert Festschrift (1922) 423–451 [Math. Zeitschr12 (1921), 219–247].

  30. P. N. 4, 179.

  31. We usea momentarily for an indicialA (i. e. an absolutec). We may plainly supposea<1.

  32. It may be shown that no other choice ofs ands′ leads to a better value ofs+s′.

  33. This is the special case of (6.31) in whicht=0.

  34. Valid since 15>8+2k.

  35. See P. N. 4, 184, for an analysis of the exceptional cases.

  36. In fact, in the notation of § 7,a s =1 (s≧k).

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Authors and Affiliations

  1. Oxford

    G. H. Hardy

  2. Cambridge

    J. E. Littlewood

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  1. G. H. Hardy
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G. H. Hardy and J. E. Littlewood, Some problems of “Partitio Numerorum”: (I) A new solution of Waring's Problem (Göttinger Nachrichten, 1920, 33–54); (II) Proof that every large number is the sum of at most 21 biquadrates (Mathematische Zeitschrift9 (1921), 14–27); (IV) The singular series in Waring's Problem, and the value of the numberG (k) (ibidem12 (1922), 161–188). We refer to these memoirs as P. N. 1, P. N. 2, P. N. 4.

We shall also have occasion to refer to the fifth memoir (P. N. 5), viz. ‘A further contribution to the study of Goldbach's Problem (Proc. London Math. Soc. (2)22 (1923), 46–56). This memoir, though concerned with a different problem, contains, in a different setting, several of the essential ideas of our present analysis.

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Hardy, G.H., Littlewood, J.E. Some problems of ‘Partitio numerorum’ (VI): Further researches in Waring's Problem. Math Z 23, 1–37 (1925). https://doi.org/10.1007/BF01506218

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