Acta Mathematica

, Volume 37, Issue 1, pp 193–239 | Cite as

Some problems of diophantine approximation

Part II. The trigonometrical series associated with the elliptic ϑ-functions
  • G. H. Hardy
  • J. E. Littlewood


Continue Fraction Trigonometrical Series Constant Multiple Diophantine Approximation Favourable Case 
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  1. 1.
    The notation is that ofTannery andMolk's Théorie des fonctions elliptiques. We shall refer to this book asT. and M. Google Scholar
  2. 1.
    Some of the properties in question are stated shortly in our paper ‘Some problems of Diophantine Approximation’ published in theProceedings of the fifth International Congress of Mathematicians, Cambridge, 1912.Google Scholar
  3. 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. 285et seq. See alsoRiemann,Werke, p. 249;Genocchi,Atti di Torino, vol. 10, p. 985.Google Scholar
  4. 1.
    Comptes Rendus, 23 Dec. 1912.Google Scholar
  5. 2.
    Hardy,Proc. Lond. Math. Soc., vol. 12, p. 370. The theorem was also discovered independently byM. Riesz.Google Scholar
  6. 1.
    The argument may even be extended to series of the type\(\sum e^{\lambda _n ix} \), where λn is not necessarily a multiple of π; but for this we require a whole series of theorems concerningDirichlet's series.Google Scholar
  7. 2.
    The formula is due toGenocchi andSchaar. SeeLindelöf,l. c., p. 75, for references to the history of the formula.Google Scholar
  8. 1.
    TheA in this formula is of course not thesame numerical constant as before.Google Scholar
  9. 1.
    It is these facts which render necessary the analysis of 2. 124.Google Scholar
  10. 1.
    Lindelöf,l.c., p. 44.Google Scholar
  11. 1.
    Hardy,Orders of Infinity, p. 17.Google Scholar
  12. 2.
    SeeBorel,Rendiconti di Palermo, Vol. 27, p. 247, andMath. Annalen, Vol. 72, p. 578;Bernstein,Math. Annalen, Vol. 71, p. 417 and Vol. 72, p. 585.Google Scholar
  13. 1.
    Hardy andLittlewood,Proc. Lond. Math. Soc., Vol. 11, p. 433.Google Scholar
  14. 1.
    Hardy,Quarterly Journal, Vol. 44, p. 147.Google Scholar
  15. 2.
    Hardy,l. c. Quarterly Journal, Vol. 44, p. 156.Google Scholar
  16. 1.
    Hardy,l. c. Quarterly Journal, Vol. 44, p. 150.Google Scholar
  17. 2.
    Or along any ‘regular path’ which does not touch the circle of convergence.Google Scholar
  18. 1.
    T. and M., Vol. 2, p. 262 (Table XLII).Google Scholar
  19. 1.
    The formulaf≈ϕ implies that ∣f∣/ϕ lies between fixed positive limits: seeHardy,Orders of Infinity, pp. 2, 5.Google Scholar
  20. 1.
    Hardy andLittlewood,Proc. Lond. Math. Soc., Vol. 11, p. 435.Google Scholar
  21. 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 theRecords of Proceedings of the London Math. Soc. for 13 Febr. 1913.Google Scholar
  22. 2.
    Acta Mathematica, Vol. 30, p. 398.Google Scholar
  23. 3.
    Rendiconti di Palermo, Vol. 32, p. 386.Google Scholar
  24. 4.
    The cosine series converges whenx is a rational of the form (2λ+1)/(2μ+1) or 2λ(4μ+3), the sine series whenx 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
  25. 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, forno values ofx. 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
  26. 1.
    Math. Annalen, Vol. 61, p. 251. See alsoLeçons 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 replacen −α byn −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
  27. 3.
    This is the ‘Riesz-Fischer Theorem’.Google Scholar
  28. 4.
    W. H. Young,Proc. Lond. Math. Soc., Vol. 12, p. 71.Google Scholar

Copyright information

© Almqvist & Wiksells Boktryckeri-A.-B. 1914

Authors and Affiliations

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

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