1.

*Math Annalen*, vol. 57, 1903, pp. 195–204;Landau,*Handbuch*, pp. 711*et seq*. Naturally our argument does not give so large a value of*K* asSchmidt's. The actual inequalities proved bySchmidt are not the inequalities (1. 143) but the substantially equivalent inequalities (1. 51).

1.

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.

1.

Landau,*Hadbuch* p. 816

2.

*Acta Mathematica*, vol. 40, 1916, pp. 185–190.

3.

*Math. Annalen*, vol. 71, 1912, pp. 548–564

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).

1.

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.

CrossRefMATH2.

*Comptes Rendus*, 6 April, 1914.

3.

*Math. Annalen*, vol. 76, 1915, pp. 212–243.

1.

SeeLandau,*Handbuch*, pp. 401*et seq*.

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).

3.

*Comptes Rendus*, 22 June 1914.

1.

See the references inLandau's bibliography, andLehmer's
*List of prime numbers from 1 to 10,006,721* (Washington, 1914).

2.

Bohr andLandau,*Göttinger Nachrichten*, 1910, pp. 303–330.

3.

*Comptes Rendus*, 29 Jan. 1912.

1.

*Math. Annalen*, vol. 74, 1913, pp. 3–30.

2.

Compare,

Landau,

*Math. Annalen*, vol. 61, 1905, pp. 527–550.

CrossRefMATH1.

SeeLandau,*Prace Matematyczno Fizyczne*, vol. 21, p. 170.

1.

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.

1.

The argument is so much like that ofLandau (*Prace Matematyczno-Fizyczne*, vol. 21, pp. 173*et 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 that*T*→∞.

1.

*Handbuch*, p. 874.

2.

*l. c. Handbuch*, pp. 128, 130 (pp. 173*et scq.*)

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's*Handbuch* (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 when*t*→∞.

1.

Landau,*Handbuch*, p. 336.

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*|<*2*π, is of course evident.

1.

2.

*Handbuch*, pp. 337*et 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).

1.

In our paper ‘Some Problems of Diophantine Approximation’,*Acta Mathematica*, vol. 37, p. 225, we defined*f*=*Q*(ϕ) as meaning*f*≠*o*(ϕ). The notation adopted here is a natural extension.

1.

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*).

CrossRefMATH 1.

M. Riesz,*Comptes Rendus*, 5 July and 22 Nov. 1909.

2.

M. Riesz,*Comptes Rendus*, 12 June 1911.

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.

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.

5.

It σ is an integer, then*S*(I/ω) is a finite series which may include logarithms. It is in any case without importance.

1.

See I. 2.

1.

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.

CrossRefMATHMathSciNet2.

The ‘trivial’ zeros of*L(s)* are*s*=−1, −3, −5, ...: seeLandau,*Handbuch*, p. 498.\(\Phi \left( y \right) = \Phi _1 \left( y \right) + y\log \left( {\frac{1}{y}} \right)\Phi _2 \left( y \right).\)

1.

Our argument is modelled on one applied to the Zeta-function byJensen,*Comptes Rendus*, 25 april 1887.

1.

It is fact true that ϒ_{1} > 6 seeGrossmann,*Dissertation*, Göttingen, 1913.

1.

Cf.W. H. Young,*Proc. London Math. Soc.*, ser. 2, vol. 12, pp. 41–70.

2.

We suppose that*a*
_{1}=0,*a*
_{1}=0, as evidently we may do without loss of generality.

1.

See the footnote to p. 140.

1.

SeeLandau,*Handbuch*, p. 816.

2.

Using the functional equation.

1.

Whittaker, andWatson,*Modern Analysis* et. 2, pp. 367, 377.

1.

These transformations are the same as those used byHardy,*Comptes Rendus*, 6 April 1914.

1.

In forming the series of residues we have assumed, for simplicity, that the poles are all simple.

1.

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.

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.

2.

Mellin,

*Acta mathematica*, vol. 25, 1902, pp. 139–164, 165–184 (p.159): see also

Nielsen,

*Handbuch der Theorie der Gamma-Funktion*, pp. 221

*et seq*.

CrossRef 1.

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.

CrossRef2.

Comptes Rendus, 29 Jan. 1912.

1.

*Math. Annalen*, vol. 71, 1912, pp. 548–564.

1.

Landau,*Handbuch*, p. 336.

1.

Observing that\(\frac{I}{x}< \frac{I}{{x_0 }}\), where*x*
_{0}=θ_{0}^{a}, and that log*xT*
^{a}>*a*log*T*+log*x*
_{0}.

2.

Landau,*Handbuch*, p. 339.

1.

Cf. Clandau,*Math. Annalen*, vol. 71, 1912, p. 557.

1.

Landau,*Handbuch*, p. 8c6.

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).

2.

The terms have to be retained in*J*
_{2} because ε/e, though outside the range of integration, may be very near to one of the limits.

1.

See section 1 for a summary of previous results.

1.

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).

CrossRefMATH1.

Landau,*Handbuch*, p. 868.

1.

Landau,*l. c. supra Handbuch*, p. 868.

1.

Landau,*Handbuch*, p. 806.

2.

Landau,*l.c. supra Handbuch*, p. 806.

1.

Landau,*Handbuch*, pp. 712*et seq*.

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), the*O* in this formula and the corresponding*O* in (5. 121) can each be replaced by*o*.

2.

See pp. 387, 351.

1.

SeeBohr andLandau,*Göttinger Nachrichten*, 1910, pp. 303–330, and a number of later papers byBohr.

2.

The notation is that of our first paper, ‘Some problems of Diophantine Approximation’,*Acta Mathematica*, vol. 37, pp. 155–193.

1.

*Göttinger Nachrichten*, 1910, p. 316.