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Literatur
Tannery andMolk, Éléments de la théorie des Fonctions Elliptiques, Vol. II (1896), p. 257. We shall refer to this book (Vol. II) as F.E.
Cf.Hardy, On the representation of a number as the sum of any number of squares, and in particular of five,Transactions of the American Mathematical Society, Vol. XXI (1920) pp. 255–284 (p. 259). Though the direct method gives the result in many cases without necessitating an appeal to the transformation theory, the latter has the advantage of being applicable to all elliptic modular functions, including those for which a direct method is not available. SeeHardy andRamanujan, Asymptotic formulae in combinatory analysis,Proceedings of the London Mathematical Society, (Ser. 2) Vol. 17 (1918) pp. 75–115 (pp. 93, 94).
F. E. p. 262.
To avoid constant repetition we shall understand that throughout this paper the path along which τ tends to ξ is the straight linex=ξ.
Hardy andLittlewood, Some problems of Diophantine approximation (II),Acta Mathematica, Vol. 37, (1914), pp. 193–238, (pp. 226–230).
F.E. pp. 262, 266, 267.
In other words,g is the least common multiple of 2 andr.
The results of Lemma 1 were given in a somewhat different form in my, paper, Some Diophantine approximations connected with quadratic surds,Journal of the Indian Mathematical Society, Vol. XIV (1922), pp. 161–166.
Whenr=1 there is only one ϑ and one φ. ϑ=[0,a 1,a 1,a 1,...], φ=[a 1,a 1,...].
See, for example,Chrystal, Algebra, Vol. II (1922), p. 433.
Cf.Hardy andLittlewood, loc. cit., p. 229.
F. E. pp. 109–111.
Owing to the cyclic order in which thea's appear as partial quotients in the continued fraction, one may regard (in the notation given in Lemma 1) a period as beginning with anyd t (t≤t≤r) and ending withd t+r−1 . The convention adopted here is necessary to make our definitions, that follow, unambiguous.
The sequence formed by the residues ofp 1,p 2,... to any fixed modulusM (the residues lying between 1 andM) is also periodic. The proof is the same as that for the caseM=8 given above.
In order to show clearly the contents of the proof it is supposed here thatr≥4. The formal alterations necessary whenr<4 can be easily seen. Whenr=1 it will be convenient to regard the period as consisting of two equal partial quotients; this is clearly permissible.
The periods of the p- and the q-sequences are, of course, not necessarily the same. In the course of the proof of Lemma 3 the period whose existence is proved is a multiple ofr. But smaller periods may very well exist, which are not divisible byr. Thus, for example, if everya is a multiple of 8, it is easily seen that 2 is a period.
F.E., p. 109.
F.E. p. 262, formula (3) withv=V=0. I have slightly altered the notation and interchanged τ andT.
F.E. p. 91.
F.E. p. 262 formulae (5) and (7).
F.E. p. 241 Table (6), and, p. 262 Table (8).
This implies that the ω-sequence is periodic with periodH.
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Ananda-rau, K. On the boundary behaviour of elliptic modular functions. Acta Math. 52, 143–168 (1929). https://doi.org/10.1007/BF02592684
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DOI: https://doi.org/10.1007/BF02592684