Contributions to the theory of the Riemann Zeta-function and the theory of the distribution of primes. (English) JFM 46.0498.01

Hardy, G. H.; Littlewood, J. E.

Acta Math. 41, 119-196 (1917).

Eine Arbeit von ganz ungewöhnlichem Reichtum des Inhalts! Es ist unmöglich, die Fülle der darin enthaltenen Gedanken in einer kurzen Besprechung anzudeuten. Viele von ihnen sind schon in früheren verstreuten Noten der Verff. ausgesprochen (vgl. die unten gegeben Zitate), von denen nun hier eine zusammenfassende Darstellung gegeben wird. Wir müssen uns mit der Angabe einiger Hauptsachen begnügen.
In Abschn. 1 werden mit Hilfe des Cahen-Mellinschen Integrals \[ \frac{1}{2\pi i}\int_{\kappa- i\infty}^{\kappa+i\infty}\Gamma(s)y^{-s}\,ds=e^{-y}\quad (\kappa>0, \text{Re}(y)>0), \] und in Verbindung mit gewissen Reihensätzen “Tauberscher Art” die die Verf. früher veröffentlicht haben (s. etwa [Proc. Lond. Math. Soc. (2) 11, 411–478 (1912; JFM 43.0312.01), Messenger (2) 42, 191–192 (1913; JFM 44.0283.01)]) sehr weitgehende Konvergenzsätze über allgemeine Dirichletsche Reihen bewiesen, die als einfache z. B. den Primzahlsatz und viele äquivalente Sätze liefern.
In Abschn. 2 wird die Funktion \[ F(y)=\sum(\Lambda(n)-1)e^{-ny}, \quad\text{Re}(y)>0 \]
für \(y\to 0\) untersucht. \([\Lambda(n)=\log p\) für \(n=p^m\), sonst 0.] Unter der Annahme der Richtigkeit der Riemannschen Vermutung wird bewiesen, daß \(F(y)=O(y^{-\frac 12})\) ist, während andrerseits ein \(K>0\) existiert, so daß in jeder Nähe von 0 \[ F(y)<-Ky^{-\frac 12},+Ky^{-\frac 12}<F(y) \]
gilt. [Hierin steckt als Korollar ein ähnlicher von E. Schmidt herrührender Satz über \(\psi(x)-x\); mit \(\psi(x)=\sum_{n\leqq x} \Lambda(n)\)].
Ähnliche Sätze über die Funktion \[ f(y)=\sum\mu(n)e^{-ny} \]
gelingt es noch nicht zu beweisen.
3. Abschn. handelt von der Tschebyscheffschen Vermutung, von der in einem früheren Referat einer Landauschen Arbeit genauer die Rede war. (Vgl. E. Landau, Gött. Nachr. 1917, 96–101 (1917; JFM 46.0264.01)].)
Im 4. Abschn. wird für \(T\to+\infty\) die asymptotische Abschätzung \[ \int_{-T}^{+T} | \zeta(\frac 12+it)|^2\,dt \sim 2T\log T \]
bewiesen, der bisher erst analoge Formeln für die Geraden \(\text{Re}(s)=\beta\gtrless\frac 12\) vorangingen.
Der 5. Abschn. bringt Formeln von ähnlichem Charakter, wie die in der vorstehenden Note von M. Riesz [Acta Math. 40, 185–190 (1916; JFM 46.0497.03)] aufgestellte.
Diese fünf Abschnitte bilden ein erstes Kapitel. Im II. Kap. handelt es sich um die Reihe \[ \sum_\rho e^{\alpha\rho\log(-i\rho)}x^\rho \rho^{-\kappa}, \] in der \(a,x\) und \(\kappa\) reell sind und die Summation sich auf die komplexen Nullstellen \(\rho\) von \(\zeta(s)\) bezieht. E. Landau hatte sich [Math. Ann. 71, 548–564 (1912; JFM 43.0332.01)] mit der spezielleren Reihe \(\sum_\rho x^\rho \rho^{-\nu}\) beschäftigt. Sätze, die den von Landau bewiesenen ähnlich sind, werden jetzt über die genannte allgemeinere Reihe aufgestellt.
Das III. Kap. beschäftigt sich mit den auf der Geraden \(\text{Re}(s)=\frac 12\) gelegenen Nullstellen der \(\zeta\)- Funktion, von denen G. H. Hardy unendlich viele nachgewiesen hatte [C. R. 158, 1012–1014 (1914; JFM 45.0716.04)]. Deren Anzahl wird nach unten abgeschätzt mit einer Schärfe, die über alles bisher bekannte weit hinausgeht: die Anzahl derselben, deren Ordinate zwischen 1 und \(T\) liegt, ist für kein positives \(\delta\) gleich \(o(T^{\frac 34-\delta})\). Die Versuche der Verf., das Entsprechende für \(o(T^{1-\delta})\) zu beweisen, haben noch keinen Erfolg gehabt.
Das letzte Kapitel endlich enthält eine ausführliche Darstellung der Resultate, die (ohne Beweis) schon von J. E. Littlewood [C. R. 158, 1869–1872 (1914)] veröffentlicht waren und über die in JFM 45.0305.01 berichtet worden war.

Reviewer: Knopp, Prof. (Königsberg)

MSC:

11M06	\(\zeta (s)\) and \(L(s, \chi)\)
11N05	Distribution of primes

JFM Section:

Vierter Abschnitt. Analysis. Kapitel 4. Allgemeine Theorie der Funktionen komplexer Argumente. Grundlagen und Allgemeines. Potenzreihen. Dirichletsche Reihen. Fakultätenreihen und Verwandtes. Ganze transzendente Funktionen. Andere Klassen von Funktionen. Folgen von Funktionen.

Citations:

JFM 43.0312.01; JFM 44.0283.01; JFM 46.0264.01; JFM 46.0497.03; JFM 43.0332.01; JFM 45.0716.04; JFM 45.0305.01

References:

[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).
[2]	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.
[3]	Landau,Hadbuch p. 816
[4]	Acta Mathematica, vol. 40, 1916, pp. 185–190.
[5]	Math. Annalen, vol. 71, 1912, pp. 548–564
[6]	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).
[7]	SeeGram,Acta Mathematica, vol. 27, 1903, pp. 289–304;Lindelöf,Acta Societatis Fennicoe, 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. · JFM 34.0461.01 · doi:10.1007/BF02421310
[8]	Comptes Rendus, 6 April, 1914.
[9]	Math. Annalen, vol. 76, 1915, pp. 212–243.
[10]	SeeLandau,Handbuch, pp. 401et seq.
[11]	For an explanation of this notation see our paper ’Some Problems of Diophantine Approximation (II)’,Acta Mathematica, vol. 37, pp. 193–238 (p. 225).
[12]	Comptes Rendus, 22 June 1914.
[13]	See the references inLandau’s bibliography, andLehmer’s List of prime numbers from 1 to 10,006,721 (Washington, 1914).
[14]	Bohr andLandau,Göttinger Nachrichten, 1910, pp. 303–330.
[15]	Comptes Rendus, 29 Jan. 1912.
[16]	Math. Annalen, vol. 74, 1913, pp. 3–30.
[17]	Compare,Landau,Math. Annalen, vol. 61, 1905, pp. 527–550. · JFM 37.0224.04 · doi:10.1007/BF01449495
[18]	SeeLandau,Prace Matematyczno Fizyczne, vol. 21, p. 170.
[19]	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)y 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.
[20]	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.
[21]	Handbuch, p. 874.
[22]	l. c. Handbuch, pp. 128, 130 (pp. 173et scq.)
[23]	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 |{\(\Gamma\)}({\(\sigma\)}+ti)y {\(\sigma\)}+ti |, where (ity)>o, tends to zero like an exponential whent.
[24]	Landau,Handbuch, p. 336.
[25]	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 (y) is a power-series convergent for |y|<2{\(\pi\)}, is of course evident.
[26]	Gram,l. c.. · JFM 34.0461.01 · doi:10.1007/BF02421310
[27]	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).
[28]	In our paper ’Some Problems of Diophantine Approximation’,Acta Mathematica, vol. 37, p. 225, we definedf=Q() as meaningf(). The notation adopted here is a natural extension.
[29]	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). · JFM 34.0230.02 · doi:10.1007/BF01444344
[30]	M. Riesz,Comptes Rendus, 5 July and 22 Nov. 1909.
[31]	M. Riesz,Comptes Rendus, 12 June 1911.
[32]	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.
[33]	See 2.21 for our justification of the omission of the details of the proof. Here again the integrals which occur are absolutely convergent.
[34]	It {\(\sigma\)} is an integer, thenS(I/{\(\omega\)}) is a finite series which may include logarithms. It is in any case without importance.
[35]	See I. 2.
[36]	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 {\(\sigma\)}=1/2. · JFM 45.0717.01 · doi:10.1007/BF01458139
[37]	The ’trivial’ zeros ofL(s) ares=, , , ...: seeLandau,Handbuch, p. 498. \(\Phi \left( y \right) = \Phi _1 \left( y \right) + y\log \left( {\frac{1}{y}} \right)\Phi _2 \left( y \right).\)
[38]	Our argument is modelled on one applied to the Zeta-function byJensen,Comptes Rendus, 25 april 1887.
[39]	It is fact true that 1 > 6 seeGrossmann,Dissertation, Göttingen, 1913.
[40]	Cf.W. H. Young,Proc. London Math. Soc., ser. 2, vol. 12, pp. 41–70.
[41]	We suppose thata 1=0,a 1=0, as evidently we may do without loss of generality.
[42]	See the footnote to p. 140.
[43]	SeeLandau,Handbuch, p. 816.
[44]	Using the functional equation.
[45]	Whittaker, andWatson,Modern Analysis et. 2, pp. 367, 377.
[46]	These transformations are the same as those used byHardy,Comptes Rendus, 6 April 1914.
[47]	In forming the series of residues we have assumed, for simplicity, that the poles are all simple.
[48]	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.
[49]	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.
[50]	Mellin,Acta mathematica, vol. 25, 1902, pp. 139–164, 165–184 (p.159): see alsoNielsen,Handbuch der Theorie der Gamma-Funktion, pp. 221et seq. · JFM 32.0348.02 · doi:10.1007/BF02419024
[51]	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. · JFM 46.0497.03 · doi:10.1007/BF02418544
[52]	Comptes Rendus, 29 Jan. 1912.
[53]	Math. Annalen, vol. 71, 1912, pp. 548–564.
[54]	Landau,Handbuch, p. 336.
[55]	Observing that \(\frac{I}{x}< \frac{I}{{x_0 }}\) , wherex 0={\(\theta\)} 0 a , and that logxT a >alogT+logx 0.
[56]	Landau,Handbuch, p. 339.
[57]	Cf. Clandau,Math. Annalen, vol. 71, 1912, p. 557.
[58]	Landau,Handbuch, p. 8c6.
[59]	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).
[60]	The terms have to be retained inJ 2 because {\(\epsilon\)}/e, though outside the range of integration, may be very near to one of the limits.
[61]	See section 1 for a summary of previous results.
[62]	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). · JFM 45.0717.01 · doi:10.1007/BF01458139
[63]	Landau,Handbuch, p. 868.
[64]	Landau,l. c. supra Handbuch, p. 868.
[65]	Landau,Handbuch, p. 806.
[66]	Landau,l.c. supra Handbuch, p. 806.
[67]	Landau,Handbuch, pp. 712et seq.
[68]	It has been shown byBohr, Landau, andLittlewood ({”Sur la fonction {\(\xi\)}(s) dans le voisinage de la droite {\(\sigma\)}=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.}}
[69]	See pp. 387, 351.
[70]	SeeBohr andLandau,Göttinger Nachrichten, 1910, pp. 303–330, and a number of later papers byBohr.
[71]	The notation is that of our first paper, ’Some problems of Diophantine Approximation’,Acta Mathematica, vol. 37, pp. 155–193.
[72]	Göttinger Nachrichten, 1910, p. 316.