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

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: II. Proof that every large number is the sum of at most 21 biquadrates, Mathematische Zeitschrift9 (1921), S. 14–27. The third memoir of the series (Some problems of ‘Partitio Numerorum’: III. On the expression of a number as a sum of primes) will appear shortly in the Acta Mathematica. The problems considered in this memoir are of a somewhat different character. We refer to these memoirs as P. N. 1, P. N. 2, P. N. 3.

2. D. Hilbert, Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahln-ter Potenzen, Göttinger Nachrichten 1909, S. 17–36: reprinted with certain changes in Mathematische Annalen,67 (1909), S. 281–300.

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

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

5. 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.

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6. The names are those of the authors who found the actual numbers quoted. The proofs of ‘Waring's Theorem’ for the cases in question are due to Lagrange, Maillet, Liouville, Maillet, Fleck, Wieferich, and Hurwitz (and Maillet) respectively. For detailed references see A. J. Kempner, Über das Waringsche Problem und einige Verallgemeinerungen. Inaugural-Dissertation, Göttingen 1912, and W. S. Baer, Beiträge zum Waringschen Problem, Inaugural-Dissertation, Göttingen 1913. The numbers fork=7 andk=8 could no doubt be substantially reduced. Proofs of the existence ofg(k), from which an upper bound forg(k) could be calculated, have also been given fork=10 (I. Schur),k=12 (Kempner) andk=14 (Kempner).

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

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8. Sec Kempner, loc.cit Über das Waringsche Problem und einige Verallgemeinerungen. Inaugural-Dissertation, Göttingen, S. 44–45.

9. Waring asserts quite explicitly, not merely thatg(k) exists, but thatg(2)=4,g(3)=9,g(4)=19, ‘et sic deinceps’. Nothing is known, so far as we are aware, inconsistent with the view that the numbers in question are the actual values ofg(k) for everyk.

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

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11. E. Maillet, Sur la décomposition d'un entier en une somme de puissances huitièmes d'entiers, Bulletin de la société mathématique de France,36 (1908), p. 69–77.

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

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13. See P. N. 2, S. 22 (f. n. 7). This will also appear incidentally later (S. 369), footnote

14. What we do is, in effect, to develop from our own point of view certain portions of the theory of the division of the circle (Kreisteilung). It is not unlikely that the substance of our analysis is to be found elsewhere; but it is not altogether easy to extract, from the classical accounts of the theory, the particular parts which we require.

15. A systematic account of the theory will be found in Landau'sHandbuch,1 (Zweites Buch)

16. Landau, Zur Hardy Littlewoodschen Lösung des Waringschen Problems, Göttinger Nachrichten 1921, S. 401–402.

17. Landau.Handbuch, S. 394.

18. This has been proved already, in a different manner, in P. N. 2, S. 19–21; but it is interesting to see how the result arises from our present point of view.

19. Since δ|π−1 when π+κ.

20. Landau,Handbuch, S. 479.

21. It is −1 if ξ is the principal character, and the product of a ξ and a τ if ξ is non-principal (and so primitive: Landau,Handbuch, S. 480).

22. N(16, 15)=8¹⁵ whens=15, since eachx may have any one of the values 1, 3, 5, ..., 15.

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