Acta Mathematica

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

G. H. Hardy and J. E. Littlewood

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Source
Acta Math., Volume 44 (1923), 1-70.

Dates
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485887559

Digital Object Identifier
doi:10.1007/BF02403921

Mathematical Reviews number (MathSciNet)
MR1555183

Zentralblatt MATH identifier
48.0143.04

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

Citation

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 (1923), 1--70. doi:10.1007/BF02403921. https://projecteuclid.org/euclid.acta/1485887559


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References

  • 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 the Jahresbericht der Deutschen Math.-Vereinigung, vol. 21 (1912), pp. 208–228.
  • We give here a complete list of memoirs concenred with the various applications of this method.
  • ‘Asymptotic formulae in combinatory analysis’, Comptes rendus du quatrième Congrès des mathematiciens Scandinaves à Stockholm, 1916, pp. 45–53.
  • , ‘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.
  • , ‘Some famous problems of the Theory of Numbers, and in particular Waring's Problem’, (Oxford, Clarendon Press, 1920, pp. 1–34).
  • , ‘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.
  • , ‘Note on Ramanujan's trigonometrical sum cq(n)’, Proceedings of the Cambridge Philosophical Society, vol. 20 (1921), pp. 263–271.
  • G. H. Hardy and J. E. Littlewood, ‘A new solution of Waring's Problem’, Quarterly Journal of pure and applied mathematics, vol. 48 (1919), pp. 272–293.
  • , ‘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.
  • G. H. Hardy and J. 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.
  • ‘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.
  • G. H. Hardy and S. Ramanujan, ‘Une formule asymptotique pour le nombre des partitions de n’, Comptes rendus de l'Académie des Sciences, 2 Jan. 1917.
  • , ‘Asymptotic formulae in combinatory analysis’, Proceedings of the London Mathematical Society, ser. 2, vol. 17 (1918), pp. 75–115.
  • , ‘On the coefficients in the expansions of certain modular functions’, Proceedings of the Royal Society of London (A) vol. 95 (1918), pp. 144–155.
  • E. Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 88–92.
  • 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.
  • A. Ostrowski, ‘Bemerkungen zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Mathematische Zeitschrift, vol. 9 (1921), pp. 28–34.
  • 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.
  • N. M. Shah and B. M. Wilson, ‘On an empirical formula connected with Goldbach's Theorem’, Proceedings of the Cambridge Philosophical Society, vol. 19 (1919), pp. 238–244.
  • The hypothesis must be stated in this way because (a) it has not been proved that no L(s) has real zeros between 1/2 and 1, (b) the L-functions associated with imprimitive (uneigentlich) characters have zeros on the line σ=o.
  • Naturally many of the results stated incidentally do not depend upon the hypothesis.
  • 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 his Handbuch, unless the contrary is stated.
  • χkm=o if (m,q)>I.
  • Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921) p. 497.
  • The distinction between major and minor arcs, fundamental in our work on Waring's Problem, does not arise here.
  • Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), p. 421.
  • Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 572–573.
  • 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.
  • Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 485, 489, 492.
  • See the additional note at the end.
  • Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), pp. 509, 510, 519.
  • Landau, ‘Zur Hardy-Littlewood'schen Lösung des Waringschen Problems’, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen (1921), p. 511 (footnote).
  • 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, since Y−3 Д(s) tends to zero, when |t|↦∞, like an exponential e−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.
  • 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).
  • 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).
  • 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).
  • 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).
  • 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).
  • 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).
  • Σ refers to the complex zeros of L1(s), not merely to those of ζ(s)
  • 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).
  • The argument fails if q=1, or q=2; but c1 (n)=c1(−n)=1, c2(n)=c2(−n)=−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).
  • As regards the earlier history of ‘Goldbach's Theorem’, see L. E. Dickson, History of the Theory of Numbers, vol. 1 (Washington 1919), pp. 421–425.
  • 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.
  • Landau, p. 218. E. Landau, ‘Gelöste
  • P. Stäckel, ‘Über Goldbach's empirisches Theorem: Jede grade Zahl kann als Summe von zwei Primzahlen dargestellt werden’, Göttinger Nachrichten, 1896, pp. 292–299.
  • E. Landau, ‘Über die zahlentheoretische Funktion ϕ(n) und ihre Beziehung zum Goldbachschen Satz’, Göttinger Nachrichten, 1900, pp. 177–186.
  • J. Merlin, ‘Un travail sur les nombres premiers,’, Bulletin des sciences mathématiques, vol. 39 (1915), pp. 121–136.
  • 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 forme ap+b’, ibid. part. 4 (1917). no. 14, pp. 1–9; and the postscript to this memoir.
  • P. Stäckel, ‘Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen’, 8 August 1916; ‘Die Lückenzahlen r-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.
  • Throughout 4. 2A is the same constant.
  • For general theorems including those used here as very special cases, see K. Knopp, Divergenzcharactere gewisser Dirichlet'scher Reihen’, Acta Mathematica, vol. 34, 1909, pp. 165–204 (e. g. Satz III, p. 176).
  • Landau, p. 218. E. Landau, ‘Gelöste
  • Whether Sylvester's argument was or was not we have no direct means of judging.
  • Probability is not a notion of puro mathematies, but of philosophy or physics.
  • Compare Shah and Wilson, l. c.) p. 238. The same conclusion may be arrived at in other ways.
  • , p. 242.
  • 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.
  • Note that S2=o if k is odd, as it should be.
  • 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.
  • 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.
  • The fourth was that of the existence of a prime between n2 and (n+1)2 for every n>0. The problem of primes am2+bm+c must not be confused with the much simpler (though still difficult) problem of primes included in the definite quadratic form ax2+bxy+cy2 in two independent variables. This problem, of course, was solved in the classical researches of de 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.
  • Even this is a formal process, for (5. 412) is not absolutely convergent.
  • See Dirichlet-Dedekind, Vorlesungen über Zahleutheorie, ed. 4 (1894), pp. 293et seq.
  • By Stern 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 primes q=4m+3 into sums q=p+2p′, where p and p′ are primes of the form 4m+1, extending as far as q=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).
  • 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).
  • 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).
  • It is here that we use the condition ar┼as.
  • To avoid any possible misunderstanding, we repeat that these theorems are consequences of Hypothesis X.
  • L. E. Dickson, History of the Theory of Numbers, vol. I, p. 355.