# Some problems of diophantine approximation

Part II. The trigonometrical series associated with the elliptic ϑ-functions

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## Keywords

Continue Fraction Trigonometrical Series Constant Multiple Diophantine Approximation Favourable Case
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## Literature

- 1.The notation is that ofTannery andMolk's
*Théorie des fonctions elliptiques*. We shall refer to this book as*T. and M.*Google Scholar - 1.Some of the properties in question are stated shortly in our paper ‘Some problems of Diophantine Approximation’ published in the
*Proceedings of the fifth International Congress of Mathematicians*, Cambridge, 1912.Google Scholar - 1.This result (or rather the analogous result for the sine series) is stated byBromwich,
*Infinite Series*, p. 485, Ex. 10. We have been unable to find any complete discussion of the question, but the necessary materials well be found inDirichlet-Dedekind,*Vorlesungen über Zahlentheorie*, pp. 285*et seq*. See alsoRiemann,*Werke*, p. 249;Genocchi,*Atti di Torino*, vol. 10, p. 985.Google Scholar - 1.
*Comptes Rendus*, 23 Dec. 1912.Google Scholar - 2.Hardy,
*Proc. Lond. Math. Soc.*, vol. 12, p. 370. The theorem was also discovered independently byM. Riesz.Google Scholar - 1.The argument may even be extended to series of the type\(\sum e^{\lambda _n ix} \), where λ
_{n}is not necessarily a multiple of π; but for this we require a whole series of theorems concerningDirichlet's series.Google Scholar - 2.The formula is due toGenocchi andSchaar. SeeLindelöf,
*l. c.*, p. 75, for references to the history of the formula.Google Scholar - 1.
- 1.It is these facts which render necessary the analysis of 2. 124.Google Scholar
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- 2.SeeBorel,
*Rendiconti di Palermo*, Vol. 27, p. 247, and*Math. Annalen*, Vol. 72, p. 578;Bernstein,*Math. Annalen*, Vol. 71, p. 417 and Vol. 72, p. 585.Google Scholar - 1.
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- 2.Or along any ‘regular path’ which does not touch the circle of convergence.Google Scholar
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*T. and M.*, Vol. 2, p. 262 (Table XLII).Google Scholar - 1.The formula
*f*≈ϕ implies that ∣*f*∣/ϕ lies between fixed positive limits: seeHardy,*Orders of Infinity*, pp. 2, 5.Google Scholar - 1.
- 1.An abstract of the contents of this part of the paper appeared, under the title, «Trigonometrical Series which Converge Nowhere or Almost Nowhere», in the
*Records of Proceedings of the London Math. Soc.*for 13 Febr. 1913.Google Scholar - 2.
*Acta Mathematica*, Vol. 30, p. 398.Google Scholar - 3.
*Rendiconti di Palermo*, Vol. 32, p. 386.Google Scholar - 4.The cosine series converges when
*x*is a rational of the form (2λ+1)/(2μ+1) or 2λ(4μ+3), the sine series when*x*is a rational of the form (2λ+1)/(2μ+1) or 2λ/(4μ+1) (see 2. 01). In the abstract referred to above this part of the result (which is of course trivial) was stated incorrectly.Google Scholar - 5.It is only since this paper was written that we have become aware of a different solution given byH. Steinhaus (
*Comptes Rendus de la Société Scientifique de Varsovie*, 1912, p. 223).Steinhaus also solves the problem of convergence for his series completely; they converge, in fact, for*no*values of*x*. Thus in this respect our examples have no advantage over his; the advantage, if anywhere, is on his side. In respect of simplicity etc. our examples have the advantage over his as much as overLusin's.Google Scholar - 1.
*Math. Annalen*, Vol. 61, p. 251. See also*Leçons sur les séries trigonométriques*, p. 94 where however the proof is inaccurate.A Fourier's series is in fact summable (*C*°), for any positive °, almost everywhere (Hardy,*Proc. Lond. Math. Soc.*, Vol. 12 p. 365). That our series are notFourier's series when α<1/2 can in fact be inferred merely from their non-convergence, since to replace*n*^{−α}by*n*^{−β}where β is any number greater than α, would, if they wereFourier's series, render them convergent almost everywhere (Young,*Comptes Rendus*, 23 Dec. 1912).Google Scholar - 3.This is the ‘Riesz-Fischer Theorem’.Google Scholar
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© Almqvist & Wiksells Boktryckeri-A.-B. 1914