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

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References

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    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.

  21. 2

    Naturally many of the results stated incidentally do not depend upon the hypothesis.

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    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.

  23. 1

    χk m=o if (m,q)>I.

  24. 2

    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921) p. 497.

  25. 1

    The distinction between major and minor arcs, fundamental in our work on Waring's Problem, does not arise here.

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    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), p. 421.

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    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 572–573.

  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.

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    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 485, 489, 492.

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    See the additional note at the end.

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    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 509, 510, 519.

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    Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’,Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), p. 511 (footnote).

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    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.

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    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).

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    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).

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    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).

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    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).

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    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).

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    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).

  40. 2

    Σ refers to the complex zeros ofL 1 (s), not merely to those of ζ(s)

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    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).

  42. 1

    The argument fails ifq=1, orq=2; butc 1 (n)=c 1(−n)=1,c 2(n)=c 2(−n)=−1.

  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).

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    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.

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    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.

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    Landau, p. 218.E. Landau, ‘Gelöste

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    P. Stäckel, ‘Über Goldbach's empirisches Theorem: Jede grade Zahl kann als Summe von zwei Primzahlen dargestellt werden’,Göttinger Nachrichten, 1896, pp. 292–299.

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    E. Landau, ‘Über die zahlentheoretische Funktion ϕ(n) und ihre Beziehung zum Goldbachschen Satz’,Göttinger Nachrichten, 1900, pp. 177–186.

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    J. Merlin, ‘Un travail sur les nombres premiers,’,Bulletin des sciences mathématiques, vol. 39 (1915), pp. 121–136.

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    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.

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    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.

  52. 2

    Throughout 4. 2A is the same constant.

  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).

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    Landau, p. 218.E. Landau, ‘Gelöste

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    Whether Sylvester's argument was or was not we have no direct means of judging.

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    Probability is not a notion of puro mathematies, but of philosophy or physics.

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    Compare Shah and Wilson,l. c.) p. 238. The same conclusion may be arrived at in other ways.

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    , p. 242.

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  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.

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    Note thatS 2 =o ifk is odd, as it should be.

  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.

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    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.

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  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.

  64. 1

    Even this is a formal process, for (5. 412) is not absolutely convergent.

  65. 2

    SeeDirichlet-Dedekind,Vorlesungen über Zahleutheorie, ed. 4 (1894), pp. 293et seq.

  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).

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    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).

  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).

  69. 1

    It is here that we use the conditiona r ┼a s .

  70. 1

    To avoid any possible misunderstanding, we repeat that these theorems areconsequences of Hypothesis X.

  71. 1

    L. E. Dickson,History of the Theory of Numbers, vol. I, p. 355.

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Hardy, G.H., Littlewood, J.E. Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta Math. 44, 1–70 (1923). https://doi.org/10.1007/BF02403921

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Keywords

  • Asymptotic Formula
  • Conjugate Problem
  • Singular Series
  • Distinct Residue
  • Partitio Numerorum