Acta Mathematica

, Volume 41, Issue 1, pp 119–196 | Cite as

Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes

  • G. H. Hardy
  • J. E. Littlewood


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  1. 1.
    Math Annalen, vol. 57, 1903, pp. 195–204;Landau,Handbuch, pp. 711et seq. Naturally our argument does not give so large a value ofK asSchmidt's. The actual inequalities proved bySchmidt are not the inequalities (1. 143) but the substantially equivalent inequalities (1. 51).Google Scholar
  2. 1.
    Tschebyschef,Bulletin de l'Acadénie Impériale des Sciences de St. Petersbourg, vol. 11, 1853, p. 208, andOeuvres, vol. 1, p. 697;Landau,Rendiconti di Palermo, vol. 24, 1907, pp. 155–156.Google Scholar
  3. 1.
    Landau,Hadbuch p. 816Google Scholar
  4. 2.
    Acta Mathematica, vol. 40, 1916, pp. 185–190.Google Scholar
  5. 3.
    Math. Annalen, vol. 71, 1912, pp. 548–564Google Scholar
  6. 4.
    The idea which dominates the critical stage of the argument is alsoLandau's, but is to be found in another of his papers (‘Über die Anzahl der Gitterpunkte in gewissen Bereichen’,Göttinger Nachrichten, 1912, pp. 687–771, especially p. 707, Hilfsatz 10).Google Scholar
  7. 1.
    SeeGram,Acta Mathematica, vol. 27, 1903, pp. 289–304;Lindelöf,Acta Societatis Fennicœ, vol. 31, 1913, no. 3;Backlund,Oversigt af Finska Vetenskap Societetens Förhandlingar, vol. 54, 1911–12, A, no. 3; and further entries under these names inLandau's bibliography.CrossRefMATHGoogle Scholar
  8. 2.
    Comptes Rendus, 6 April, 1914.Google Scholar
  9. 3.
    Math. Annalen, vol. 76, 1915, pp. 212–243.Google Scholar
  10. 1.
    SeeLandau,Handbuch, pp. 401et seq.Google Scholar
  11. 2.
    For an explanation of this notation see our paper ‘Some Problems of Diophantine Approximation (II)’,Acta Mathematica, vol. 37, pp. 193–238 (p. 225).Google Scholar
  12. 3.
    Comptes Rendus, 22 June 1914.Google Scholar
  13. 1.
    See the references inLandau's bibliography, andLehmer's List of prime numbers from 1 to 10,006,721 (Washington, 1914).Google Scholar
  14. 2.
    Bohr andLandau,Göttinger Nachrichten, 1910, pp. 303–330.Google Scholar
  15. 3.
    Comptes Rendus, 29 Jan. 1912.Google Scholar
  16. 1.
    Math. Annalen, vol. 74, 1913, pp. 3–30.Google Scholar
  17. 2.
    Compare,Landau,Math. Annalen, vol. 61, 1905, pp. 527–550.CrossRefMATHGoogle Scholar
  18. 1.
    SeeLandau,Prace Matematyczno Fizyczne, vol. 21, p. 170.Google Scholar
  19. 1.
    Vol. 43, 1914, pp. 134–147. Ifan satisfies the second form of condition (i), the seriesf(y) is necessarily convergent (absolutely) fory>0, so that the first clause of condition (ii) is tnen unnecessary. There are more general forms of this theorem, involving functions such as\(y^{ - a} \left\{ {\log \left( {\frac{1}{y}} \right)} \right\}^{a_1 } \left\{ {\log \log \left( {\frac{1}{y}} \right)} \right\}^{a_2 } \cdots \cdots .,\) which we have not troubled to work out in detail. The relationf(y)Ay −a in condition (ii) must be interpreted, in the special case whenA=0, as meaningf(y)=0(y-a); and a corresponding change must be made in the conclusion.Google Scholar
  20. 1.
    The argument is so much like that ofLandau (Prace Matematyczno-Fizyczne, vol. 21, pp. 173et seq.) that it is hardly worth while to set it out in detail. We applyCauchy's Theorem to the rectangle\(c - iT,x - iT,x + iT,c - iT,\) and then suppose thatT→∞.Google Scholar
  21. 1.
    Handbuch, p. 874.Google Scholar
  22. 2.
    l. c. Handbuch, pp. 128, 130 (pp. 173et scq.)Google Scholar
  23. 1.
    The passage from (2. 211) to (2. 212) requires in reality a difficult and delicate discussion. If we suppress this part of the proof, it is because no arguments are required which involve the slightest novelty of idea. All the materials for the proof are to be found inLandau'sHandbuch (pp. 333–368). But the problem treated there is considerably more difficult than this one, inasmuch as the integrals and series dealt with are not absolutely convergent. Here everything is absolutely convergent, since |Γ(σ+ti)y σ+ti|, where ℜ(ity)>o, tends to zero like an exponential whent→∞.Google Scholar
  24. 1.
    Landau,Handbuch, p. 336.Google Scholar
  25. 1.
    This is merely another form of the ordinary formula which definesBrrnoulli's num. bers. That\(\sum {e^{ - ny} = \frac{I}{y} + \Phi \left( y \right)} \) where ϕ(y) is a power-series convergent for |y|<2π, is of course evident.Google Scholar
  26. 1.
  27. 2.
    Handbuch, pp. 337et seq. It is known that, on theRiemann hypothesis,\(N\left( {T + I} \right) - N\left( T \right) \sim \frac{{\log T}}{{2\pi }}\)(Bohr, Landau, Littlewood,Bulletins de l'Académie Royale de Belgique, 1913, no. 12, pp. 1–35).Google Scholar
  28. 1.
    In our paper ‘Some Problems of Diophantine Approximation’,Acta Mathematica, vol. 37, p. 225, we definedf=Q(ϕ) as meaningfo(ϕ). The notation adopted here is a natural extension.Google Scholar
  29. 1.
    Schmidt,Math. Annalen, vol. 57, 1903, pp. 195–204; see alsoLandau,Handbuch, pp. 712et seq. The inequalities are stated bySchmidt andLandau in terms of II(x).CrossRefMATHGoogle Scholar
  30. 1.
    M. Riesz,Comptes Rendus, 5 July and 22 Nov. 1909.Google Scholar
  31. 2.
    M. Riesz,Comptes Rendus, 12 June 1911.Google Scholar
  32. 3.
    This formula is a special case of a general formula, due toRiesz and included as Theorem 40 in the Tract ‘The general theory of Dirichlet's series’ (Cambridge Tracts in Mathematics, no. 18, 1915) byG. H. Hardy andM. Riesz.Google Scholar
  33. 4.
    See 2.21 for our justification of the omission of the details of the proof. Here again the integrals which occur are absolutely convergent.Google Scholar
  34. 5.
    It σ is an integer, thenS(I/ω) is a finite series which may include logarithms. It is in any case without importance.Google Scholar
  35. 1.
  36. 1.
    The evidence for the truth of this hypothesis is substantiantially the same as that for the truth of theRiemann hypothesis.Landau (Math. Ann., vol. 76, 1915, pp. 212–243) has proved that there are infinitely many zeros on the line σ=1/2.CrossRefMATHMathSciNetGoogle Scholar
  37. 2.
    The ‘trivial’ zeros ofL(s) ares=−1, −3, −5, ...: seeLandau,Handbuch, p. 498.\(\Phi \left( y \right) = \Phi _1 \left( y \right) + y\log \left( {\frac{1}{y}} \right)\Phi _2 \left( y \right).\) Google Scholar
  38. 1.
    Our argument is modelled on one applied to the Zeta-function byJensen,Comptes Rendus, 25 april 1887.Google Scholar
  39. 1.
    It is fact true that ϒ1 > 6 seeGrossmann,Dissertation, Göttingen, 1913.Google Scholar
  40. 1.
    Cf.W. H. Young,Proc. London Math. Soc., ser. 2, vol. 12, pp. 41–70.Google Scholar
  41. 2.
    We suppose thata 1=0,a 1=0, as evidently we may do without loss of generality.Google Scholar
  42. 1.
    See the footnote to p. 140.Google Scholar
  43. 1.
    SeeLandau,Handbuch, p. 816.Google Scholar
  44. 2.
    Using the functional equation.Google Scholar
  45. 1.
    Whittaker, andWatson,Modern Analysis et. 2, pp. 367, 377.Google Scholar
  46. 1.
    These transformations are the same as those used byHardy,Comptes Rendus, 6 April 1914.Google Scholar
  47. 1.
    In forming the series of residues we have assumed, for simplicity, that the poles are all simple.Google Scholar
  48. 1.
    We can prove thatsome such sequence of curves as is referred to above exists, and that our series can be rendered convergent bysome process of bracketing terms: but we can prove nothing about the distribution of the curves or the size of the brackets.Google Scholar
  49. 2.
    As we do not profess to be able to give rigorous proofs of the main formulae of this sub-section, it seems hardly worth which to state such conditions in detail.Google Scholar
  50. 2.
    Mellin,Acta mathematica, vol. 25, 1902, pp. 139–164, 165–184 (p.159): see alsoNielsen,Handbuch der Theorie der Gamma-Funktion, pp. 221et seq.CrossRefGoogle Scholar
  51. 1.
    SeeRiez,Acta mathematica, vol. 40, 1916, pp. 185–190. The actual formula communicated to us byRiesz (in 1912) was not this one, nor the formula for\(\frac{I}{{\zeta \left( s \right)}}\), contained in his memoir, but the analgogous formula for\(\frac{I}{{\zeta \left( {s + I} \right)}}\). All of these formulae may be deduced fromMellin's inversion formula already referred to in 2.53. The idea of obtaining a necessary and sufficient condition of this character for the truth of theRiemann hypothesis is of courseRiesz's and not ours.CrossRefGoogle Scholar
  52. 2.
    Comptes Rendus, 29 Jan. 1912.Google Scholar
  53. 1.
    Math. Annalen, vol. 71, 1912, pp. 548–564.Google Scholar
  54. 1.
    Landau,Handbuch, p. 336.Google Scholar
  55. 1.
    Observing that\(\frac{I}{x}< \frac{I}{{x_0 }}\), wherex 00a, and that logxT a>alogT+logx 0.Google Scholar
  56. 2.
    Landau,Handbuch, p. 339.Google Scholar
  57. 1.
    Cf. Clandau,Math. Annalen, vol. 71, 1912, p. 557.Google Scholar
  58. 1.
    Landau,Handbuch, p. 8c6.Google Scholar
  59. 1.
    The fundamental idea in the analysis which follows is the same as that ofLandau's memoir ‘Über die Anzahl der Gitterpunkte in gewissen Bereichen’ (Göttinger Nachrichten, 1912, pp. 687–771).Google Scholar
  60. 2.
    The terms have to be retained inJ 2 because ε/e, though outside the range of integration, may be very near to one of the limits.Google Scholar
  61. 1.
    See section 1 for a summary of previous results.Google Scholar
  62. 1.
    The general idea used in this part of the proof is identical with that introduced byLandau in his simplification ofHardy's proof of the existence of an infinity of roots (seeLandau,Math. Annalen, vol. 76, 1915, pp. 212–243).CrossRefMATHGoogle Scholar
  63. 1.
    Landau,Handbuch, p. 868.Google Scholar
  64. 1.
    Landau,l. c. supra Handbuch, p. 868.Google Scholar
  65. 1.
    Landau,Handbuch, p. 806.Google Scholar
  66. 2.
    Landau,l.c. supra Handbuch, p. 806.Google Scholar
  67. 1.
    Landau,Handbuch, pp. 712et seq.Google Scholar
  68. 1.
    It has been shown byBohr, Landau, andLittlewood (»Sur la fonction ξ(s) dans le voisinage de la droite σ=1/2»,Bulletins de l'Académie Royale de Belgique, 1913, pp. 1144–1173) that, on theRiemann hypothesis (which we are now assuming), theO in this formula and the correspondingO in (5. 121) can each be replaced byo.Google Scholar
  69. 2.
    See pp. 387, 351.Google Scholar
  70. 1.
    SeeBohr andLandau,Göttinger Nachrichten, 1910, pp. 303–330, and a number of later papers byBohr.Google Scholar
  71. 2.
    The notation is that of our first paper, ‘Some problems of Diophantine Approximation’,Acta Mathematica, vol. 37, pp. 155–193.Google Scholar
  72. 1.
    Göttinger Nachrichten, 1910, p. 316.Google Scholar

Copyright information

© Almqvist & Wiksells Boktryckeri-A.-B. 1916

Authors and Affiliations

  • G. H. Hardy
    • 1
  • J. E. Littlewood
    • 1
  1. 1.Trinity CollegeCambridge

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