Acta Mathematica

, Volume 44, Issue 1, pp 1–70 | Cite as

Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes

  • G. H. Hardy
  • J. E. Littlewood

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References

  1. 1.
    E. Landau, ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceedings of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. 1, pp. 93–108 (p. 105). This address was reprinted in theJahresbericht der Deutschen Math.-Vereinigung, vol. 21 (1912), pp. 208–228.Google Scholar
  2. 2.
    We give here a complete list of memoirs concenred with the various applications of this method.Google Scholar
  3. 1.
    ‘Asymptotic formulae in combinatory analysis’,Comptes rendus du quatrième Congrès des mathematiciens Scandinaves à Stockholm, 1916, pp. 45–53.Google Scholar
  4. 2.
    , ‘On the expression of a number as the sum of any number of squares, and in particular of five or seven’,Proceedings of the National Academy of Sciences, vol. 4 (1918), pp. 189–193.CrossRefGoogle Scholar
  5. 3.
    , ‘Some famous problems of the Theory of Numbers, and in particular Waring's Problem’, (Oxford, Clarendon Press, 1920, pp. 1–34).Google Scholar
  6. 4.
    , ‘On the representation of a number as the sum of any number of squares, and in particular of five’,Transactions of the American Mathematical Society, vol. 21 (1920), pp. 255–284.CrossRefMathSciNetGoogle Scholar
  7. 5.
    , ‘Note on Ramanujan's trigonometrical sumc q (n)’,Proceedings of the Cambridge Philosophical Society, vol. 20 (1921), pp. 263–271.Google Scholar
  8. 1.
    G. H. Hardy andJ. E. Littlewood, ‘A new solution of Waring's Problem’,Quarterly Journal of pure and applied mathematics, vol. 48 (1919), pp. 272–293.Google Scholar
  9. 2.
    , ‘Note on Messrs. Shah and Wilson's paper entitled: On an empirical formula connected with Goldbach's Theorem’,Proceedings of the Cambridge Philosophical Society, vol. 19 (1919), pp. 245–254.Google Scholar
  10. 3.
    G. H. Hardy andJ. E. Littlewood, ‘Some problems of ‘Partitio numerorum’; I: A new solution of Waring's Problem’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1920), pp. 33–54.Google Scholar
  11. 4.
    ‘Some problems of ‘Partitio numerorum’ II: Proof that any large number is the sum of at most 21 biquadrates’,Mathematische Zeitschrift, vol. 9 (1921), pp. 14–27.CrossRefMathSciNetGoogle Scholar
  12. 1.
    G. H. Hardy andS. Ramanujan, ‘Une formule asymptotique pour le nombre des partitions den’, Comptes rendus de l'Académie des Sciences, 2 Jan. 1917.Google Scholar
  13. 2.
    , ‘Asymptotic formulae in combinatory analysis’,Proceedings of the London Mathematical Society, ser. 2, vol. 17 (1918), pp. 75–115.Google Scholar
  14. 3.
    , ‘On the coefficients in the expansions of certain modular functions’,Proceedings of the Royal Society of London (A) vol. 95 (1918), pp. 144–155.Google Scholar
  15. 1.
    E. Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 88–92.Google Scholar
  16. 1.
    L. J. Mordell, ‘On the representations of numbers as the sum of an odd number of squares’,Transactions of the Cambridge Philosophical Society, vol. 22 (1919), pp. 361–372.Google Scholar
  17. 1.
    A. Ostrowski, ‘Bemerkungen zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Mathematische Zeitschrift, vol. 9 (1921), pp. 28–34.CrossRefMathSciNetGoogle Scholar
  18. 1.
    S. Ramanujan, ‘On certain trigonometrical sums and their applications in the theory of numbers’,Transactions of the Cambridge Philosophical Society, vol. 22 (1918), pp. 259–276.Google Scholar
  19. 1.
    N. M. Shah andB. M. Wilson, ‘On an empirical formula connected with Goldbach's Theorem’,Proceedings of the Cambridge Philosophical Society, vol. 19 (1919), pp. 238–244.Google Scholar
  20. 1.
    The hypothesis must be stated in this way because (a) it has not been proved that noL(s) has real zeros between 1/2 and 1, (b) theL-functions associated withimprimitive (uneigentlich) characters have zeros on the line σ=o.Google Scholar
  21. 2.
    Naturally many of the results stated incidentally do not depend upon the hypothesis.Google Scholar
  22. 3.
    Landau ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921) p. 498. All references to ‘Landau’ are to hisHandbuch, unless the contrary is stated.Google Scholar
  23. 1.
    χk m=o if (m,q)>I.Google Scholar
  24. 2.
    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921) p. 497.Google Scholar
  25. 1.
    The distinction between major and minor arcs, fundamental in our work on Waring's Problem, does not arise here.Google Scholar
  26. 1.
    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), p. 421.Google Scholar
  27. 2.
    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 572–573.Google Scholar
  28. 1.
    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), p. 485. The result is stated there only for a primitive character, but the proof is valid also for an imprimitive character when (p, q)=1.Google Scholar
  29. 2.
    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 485, 489, 492.Google Scholar
  30. 3.
    See the additional note at the end.Google Scholar
  31. 1.
    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 509, 510, 519.Google Scholar
  32. 2.
    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), p. 511 (footnote).Google Scholar
  33. 1.
    This application of Cauchy's Theorem may be justified on the lines of the classical proof of the ‘explicit formulae’ for ψ(x) and π(x): see Landau, pp. 333–368. In this case the roof is much easier, sinceY −3 Д(s) tends to zero, when |t|↦∞, like an exponentiale −a|t| Compare pp. 134–135 of our memoir ‘Contributions to the theory of the Riemann Zeta-function and the theory of the distribution of primes’,Acta Mathematica, vol. 41 (1917), pp. 119–196.Google Scholar
  34. 2.
    Landau, p. 517. ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceeding of the fifth International Congress of Mathematicians Cambridge, 1913, vol. I, pp. 93–108 (p. 105).Google Scholar
  35. 1.
    Landau, p. 480. ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceeding of the fifth International Congress of Mathematicians, Cambridge, 1912, vol, I, pp. 93–108 (p. 105).Google Scholar
  36. 3.
    Landau, p. 507. ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceeding of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. I, pp. 93–108 (p. 105).Google Scholar
  37. 3.
    Landau, pp. 496, 497. ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceeding of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. I, pp. 93–108 (p. 105).Google Scholar
  38. 1.
    Landau, p. 337. ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceeding of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. I, pp. 93–108 (p. 105).Google Scholar
  39. 1.
    Landau, p. 423. ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceeding of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. I, pp. 93–108 (p. 105).Google Scholar
  40. 2.
    Σ refers to the complex zeros ofL 1 (s), not merely to those of ζ(s)Google Scholar
  41. 1.
    Landau, p. 217.E. Landau, ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceedings of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. 1, pp. 93–108 (p. 105).Google Scholar
  42. 1.
    The argument fails ifq=1, orq=2; butc 1 (n)=c 1(−n)=1,c 2(n)=c 2(−n)=−1.Google Scholar
  43. 1.
    Landau, p. 577.E. Landau, ‘Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion,Proceedings of the fifth International Congress of Mathematicians, Cambridge, 1912, vol. 1, pp. 93–108 (p. 105).Google Scholar
  44. 1.
    As regards the earlier history of ‘Goldbach's Theorem’, seeL. E. Dickson,History of the Theory of Numbers, vol. 1 (Washington 1919), pp. 421–425.Google Scholar
  45. 2.
    J. J. Sylvester, ‘On the partition of an even number into two primes’,Proc. London Math. Soc., ser. 1, vol. 4 (1871), pp. 4–6 (Math. Papers, vol. 2, pp. 709–711). See also ‘On the Goldbach-Euler Theorem regarding prime numbers’,Nature, vol. 55 (1896–7), pp. 196–197, 269 (Math. Papers, vol. 4, pp. 734–737). We owe our knowledge of Sylvester's notes on the subject to Mr.B. M. Wilson of Trinity College, Cambridge. See, in connection with all that follows, Shah and Wilson, I, and Hardy and Littlewood, 2.Google Scholar
  46. 1.
    Landau, p. 218.E. Landau, ‘GelösteGoogle Scholar
  47. 2.
    P. Stäckel, ‘Über Goldbach's empirisches Theorem: Jede grade Zahl kann als Summe von zwei Primzahlen dargestellt werden’,Göttinger Nachrichten, 1896, pp. 292–299.Google Scholar
  48. 3.
    E. Landau, ‘Über die zahlentheoretische Funktion ϕ(n) und ihre Beziehung zum Goldbachschen Satz’,Göttinger Nachrichten, 1900, pp. 177–186.Google Scholar
  49. 4.
    J. Merlin, ‘Un travail sur les nombres premiers,’,Bulletin des sciences mathématiques, vol. 39 (1915), pp. 121–136.Google Scholar
  50. 5.
    V. Brun, ‘Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare’,Archiv for Mathematik (Christiania), vol. 34 part 2 (1915), no. 8, pp. 1–15. The formula (4. 18) is not actually formulated by Brun: see the discussion by Shah and Wilson, 1, and Hardy and Littlewood, 2. See also a second paper by the same author, ‘Sur les nombres premiers de la formeap+b’,ibid. part. 4 (1917). no. 14, pp. 1–9; and the postscript to this memoir.Google Scholar
  51. 1.
    P. Stäckel, ‘Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen’, 8 August 1916; ‘Die Lückenzahlenr-ter Stufe und die Darstellung der geraden Zahlen als Summen und Differenzen ungerader Primzahlen’, I. Teil 27 Dezember 1917, II. Teil 19 Januar 1918, III. Teil 19 Juli 1918.Google Scholar
  52. 2.
    Throughout 4. 2A is the same constant.Google Scholar
  53. 2.
    For general theorems including those used here as very special cases, seeK. Knopp, Divergenzcharactere gewisser Dirichlet'scher Reihen’,Acta Mathematica, vol. 34, 1909, pp. 165–204 (e. g. Satz III, p. 176).CrossRefGoogle Scholar
  54. 2.
    Landau, p. 218.E. Landau, ‘GelösteGoogle Scholar
  55. 3.
    Whether Sylvester's argument was or was not we have no direct means of judging.Google Scholar
  56. 4.
    Probability is not a notion of puro mathematies, but of philosophy or physics.Google Scholar
  57. 5.
    Compare Shah and Wilson,l. c.) p. 238. The same conclusion may be arrived at in other ways.Google Scholar
  58. 1.
  59. 1.
    We appeal again here to the Tauberian theorem referred to at the end of 4. 2 (f. n. t), This time, of course, there is no question of an alternative argument.Google Scholar
  60. 2.
    Note thatS 2 =o ifk is odd, as it should be.Google Scholar
  61. 1.
    The series is of course divergent, and must be regarded as closed after a finite number of terms, with an error term of lower order than the last term retained.Google Scholar
  62. 2.
    J. W. L. Glaisher, ‘An enumeration of prime-pairs’,Messenger of Mathematics, vol. 8 (1878), pp. 28–33. Glaisher counts (1, 3) as a pair, so that his figure exceeds ours by I.Google Scholar
  63. 1.
    The fourth was that of the existence of a prime betweenn 2 and (n+1)2 for everyn>0. The problem of primesam 2 +bm+c must not be confused with the much simpler (though still difficult) problem of primes included in the definite quadratic formax 2 +bxy+cy 2 in two independent variables. This problem, of course, was solved in the classical researches ofde la Vallée Poussin. Our method naturally leads to de la Vallée Poussin's results, and the formal verification of them in this manner is not without interest. Here, however, our method is plainly not the right one, and could lead at best to a proof encumbered with an unnecessary hypothesis and far more difficult than the accepted proof.Google Scholar
  64. 1.
    Even this is a formal process, for (5. 412) is not absolutely convergent.Google Scholar
  65. 2.
    SeeDirichlet-Dedekind,Vorlesungen über Zahleutheorie, ed. 4 (1894), pp. 293et seq.Google Scholar
  66. 1.
    ByStern and his pupils in 1856. See-History (referred to on p. 32) p. 424. The tables constructed by Stern were presorved in the library of Hurwitz, and have been very kindly placed at our disposal by Mr. G. Pólya. These manuscrípts also contain a table of decompositions of primesq=4m+3 into sumsq=p+2p′, wherep andp′ are primes of the form 4m+1, extending as far asq=20983. The conjecture that such a decomposition is always possible (1 being counted as a prime) was made by Lagrange in 1775 (see Dickson, L. E. Dickson,History of the Theory of Numbers, vol. I (Washington 1919) p. 424).Google Scholar
  67. 1.
    See Landau, p. 67. ’Gelöste und ungelöste Problemeaus der Theorie der Primzahverteilung und der Riemanmschen Zetafunktion,Proceedings of the fifth Intemaltional longress of Mathematicions, Cambridge, 1912 vol, I, 93–108 (p. 105).Google Scholar
  68. 1.
    Landau, p. 140. ‘Gelöste und ungelöste Problemeaus der Theorie der Primzahrerteilung und der Riemanmschen Zetafunktion,Proceedings of the fifth Intemational longress of Mathematicious. Cambridge, 1912 vol. I, 93–108 (p. 105).Google Scholar
  69. 1.
    It is here that we use the conditiona r ┼a s.Google Scholar
  70. 1.
    To avoid any possible misunderstanding, we repeat that these theorems areconsequences of Hypothesis X.Google Scholar
  71. 1.
    L. E. Dickson,History of the Theory of Numbers, vol. I, p. 355.Google Scholar

Copyright information

© Almqvist & Wiksells Boktryckeri-A.-B. 1923

Authors and Affiliations

  • G. H. Hardy
    • 1
  • J. E. Littlewood
    • 2
  1. 1.New CollegeOxford
  2. 2.Trinity CollegeCambridge

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