Acta Mathematica

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

G. H. Hardy and J. E. Littlewood

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Some of the results of which this memoir contains the first full account have already been stated shortly and incompletely in the following notes and abstracts. G. H. Hardy: (1) ‘On the zeros of Riemann's Zeta-function’, Proc. London Math. Soc. (records of proceedings at meetings), ser. 2, vol. 13, 12, March 1914, p. xxix; (2) ‘Sur les zéros de la fonction ζ(s), de Riemann’, Comptes Rendus, 6 April 1914. J. E. Littlewood: ‘Sur la distribution des nombres premiers’, Comptes Rendus, 22 June 1914. G. H. Hardy and J. E. Littlewood: (1) ‘New proofs of the prime-number theorem and similar theorems’, Quarterly Journal, vol. 46, 1915, pp. 215–219; (2) ‘On the zeros of the Riemann Zeta-function’ and (3) ‘On an assertion of Tschebyschef’, Proc. London Math. Soc. (records etc.), ser. 2, vol. 14, 1915, p. xiv.

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Acta Math., Volume 41 (1916), 119-196.

First available in Project Euclid: 31 January 2017

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


Hardy, G. H.; Littlewood, J. E. Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes. Acta Math. 41 (1916), 119--196. doi:10.1007/BF02422942.

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  • Math Annalen, vol. 57, 1903, pp. 195–204; Landau, Handbuch, pp. 711et seq. Naturally our argument does not give so large a value of K as Schmidt's. The actual inequalities proved by Schmidt are not the inequalities (1. 143) but the substantially equivalent inequalities (1. 51).
  • Tschebyschef, Bulletin de l'Acadénie Impériale des Sciences de St. Petersbourg, vol. 11, 1853, p. 208, and Oeuvres, vol. 1, p. 697; Landau, Rendiconti di Palermo, vol. 24, 1907, pp. 155–156.
  • Landau, Hadbuch p. 816
  • Acta Mathematica, vol. 40, 1916, pp. 185–190.
  • Math. Annalen, vol. 71, 1912, pp. 548–564
  • The idea which dominates the critical stage of the argument is also Landau'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).
  • See Gram, 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 in Landau's bibliography.
  • Comptes Rendus, 6 April, 1914.
  • Math. Annalen, vol. 76, 1915, pp. 212–243.
  • See Landau, Handbuch, pp. 401 et seq.
  • For an explanation of this notation see our paper ‘Some Problems of Diophantine Approximation (II)’, Acta Mathematica, vol. 37, pp. 193–238 (p. 225).
  • Comptes Rendus, 22 June 1914.
  • See the references in Landau's bibliography, and Lehmer'sList of prime numbers from 1 to 10,006,721 (Washington, 1914).
  • Bohr and Landau, Göttinger Nachrichten, 1910, pp. 303–330.
  • Comptes Rendus, 29 Jan. 1912.
  • Math. Annalen, vol. 74, 1913, pp. 3–30.
  • Compare, Landau, Math. Annalen, vol. 61, 1905, pp. 527–550.
  • See Landau, Prace Matematyczno Fizyczne, vol. 21, p. 170.
  • Vol. 43, 1914, pp. 134–147. If an satisfies the second form of condition (i), the series f(y) is necessarily convergent (absolutely) for y>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 relation f(y)Ay−a in condition (ii) must be interpreted, in the special case when A=0, as meaning f(y)=0(y-a); and a corresponding change must be made in the conclusion.
  • The argument is so much like that of Landau (Prace Matematyczno-Fizyczne, vol. 21, pp. 173 et seq.) that it is hardly worth while to set it out in detail. We apply Cauchy's Theorem to the rectangle $c - iT,x - iT,x + iT,c - iT,$ and then suppose that T→∞.
  • Handbuch, p. 874.
  • l. c. Handbuch, pp. 128, 130 (pp. 173 et scq.)
  • 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 in Landau'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)yσ+ti|, where ℜ(ity)>o, tends to zero like an exponential when t→∞.
  • Landau, Handbuch, p. 336.
  • This is merely another form of the ordinary formula which defines Brrnoulli'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.
  • Gram, l. c..
  • Handbuch, pp. 337 et seq. It is known that, on the Riemann 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).
  • In our paper ‘Some Problems of Diophantine Approximation’, Acta Mathematica, vol. 37, p. 225, we defined f=Q(ϕ) as meaning fo(ϕ). The notation adopted here is a natural extension.
  • Schmidt, Math. Annalen, vol. 57, 1903, pp. 195–204; see also Landau, Handbuch, pp. 712 et seq. The inequalities are stated by Schmidt and Landau in terms of II(x).
  • M. Riesz, Comptes Rendus, 5 July and 22 Nov. 1909.
  • M. Riesz, Comptes Rendus, 12 June 1911.
  • This formula is a special case of a general formula, due to Riesz and included as Theorem 40 in the Tract ‘The general theory of Dirichlet's series’ (Cambridge Tracts in Mathematics, no. 18, 1915) by G. H. Hardy and M. Riesz.
  • See 2.21 for our justification of the omission of the details of the proof. Here again the integrals which occur are absolutely convergent.
  • It σ is an integer, then S(I/ω) is a finite series which may include logarithms. It is in any case without importance.
  • See I. 2.
  • The evidence for the truth of this hypothesis is substantiantially the same as that for the truth of the Riemann hypothesis. Landau (Math. Ann., vol. 76, 1915, pp. 212–243) has proved that there are infinitely many zeros on the line σ=1/2.
  • The ‘trivial’ zeros of L(s) are s=−1, −3, −5, ...: see Landau, Handbuch, p. 498. $\Phi \left( y \right) = \Phi _1 \left( y \right) + y\log \left( {\frac{1}{y}} \right)\Phi _2 \left( y \right).$
  • Our argument is modelled on one applied to the Zeta-function by Jensen, Comptes Rendus, 25 april 1887.
  • It is fact true that ϒ1 > 6 see Grossmann, Dissertation, Göttingen, 1913.
  • Cf. W. H. Young, Proc. London Math. Soc., ser. 2, vol. 12, pp. 41–70.
  • We suppose that a1=0, a1=0, as evidently we may do without loss of generality.
  • See the footnote to p. 140.
  • See Landau, Handbuch, p. 816.
  • Using the functional equation.
  • Whittaker, and Watson, Modern Analysis et. 2, pp. 367, 377.
  • These transformations are the same as those used by Hardy, Comptes Rendus, 6 April 1914.
  • In forming the series of residues we have assumed, for simplicity, that the poles are all simple.
  • We can prove that some such sequence of curves as is referred to above exists, and that our series can be rendered convergent by some process of bracketing terms: but we can prove nothing about the distribution of the curves or the size of the brackets.
  • 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.
  • Mellin, Acta mathematica, vol. 25, 1902, pp. 139–164, 165–184 (p.159): see also Nielsen, Handbuch der Theorie der Gamma-Funktion, pp. 221et seq.
  • See Riez, Acta mathematica, vol. 40, 1916, pp. 185–190. The actual formula communicated to us by Riesz (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 from Mellin'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 the Riemann hypothesis is of course Riesz's and not ours.
  • Comptes Rendus, 29 Jan. 1912.
  • Math. Annalen, vol. 71, 1912, pp. 548–564.
  • Landau, Handbuch, p. 336.
  • Observing that $\frac{I}{x}< \frac{I}{{x_0 }}$ , where x0=θ ${}_{0}^{a}$ , and that log xTa> alogT+logx0.
  • Landau, Handbuch, p. 339.
  • Cf. Clandau, Math. Annalen, vol. 71, 1912, p. 557.
  • Landau, Handbuch, p. 8c6.
  • The fundamental idea in the analysis which follows is the same as that of Landau's memoir ‘Über die Anzahl der Gitterpunkte in gewissen Bereichen’ (Göttinger Nachrichten, 1912, pp. 687–771).
  • The terms have to be retained in J2 because ε/e, though outside the range of integration, may be very near to one of the limits.
  • See section 1 for a summary of previous results.
  • The general idea used in this part of the proof is identical with that introduced by Landau in his simplification of Hardy's proof of the existence of an infinity of roots (see Landau, Math. Annalen, vol. 76, 1915, pp. 212–243).
  • Landau, Handbuch, p. 868.
  • Landau, l. c. supra Handbuch, p. 868.
  • Landau, Handbuch, p. 806.
  • Landau, l.c. supra Handbuch, p. 806.
  • Landau, Handbuch, pp. 712 et seq.
  • It has been shown by Bohr, Landau, and Littlewood (»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 the Riemann hypothesis (which we are now assuming), the O in this formula and the corresponding O in (5. 121) can each be replaced by o.
  • See pp. 387, 351.
  • See Bohr and Landau, Göttinger Nachrichten, 1910, pp. 303–330, and a number of later papers by Bohr.
  • The notation is that of our first paper, ‘Some problems of Diophantine Approximation’, Acta Mathematica, vol. 37, pp. 155–193.
  • Göttinger Nachrichten, 1910, p. 316.