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Additional info for Collected papers of G.H. Hardy; including joint papers with J.E. Littlewood and others
Ztforms of f(n) is it true’thdt, for any irrational value of 9, alzd tiny value of c1such that 0 s Q< 1, a sequencen, can be found such that It is easy to see that, when the increase of f(n) is sufficiently rapid, the result suggested will not generally be true. Thus, if f(n) = 2n, and 0 is a number which, when expressed in the binary scale, shows at least k O’s following upon every 1, it is plain that (299) < 9 + xk:, where X& is a number which can be made as small as we please by increasing k; sufficiently.
I , 23. Proof of the first step. To fix our ideas we shall show that we can ’ pass from a sequence (n,) giving’ to a sequence (m,) giving It will be clear that the argument is in reality of a perfectly general type, Suppose we are given aI11 &, pII, &, and that 0, rp, cy’,, (x’, , & , &, are lineThen by hypothesis we can find a sequence giving arly independent irrationals. the scheme Further, the set of points ($8, n:cp) has at least one limiting-point (A, [L), and, by restricting ourselves to a subsequence of &), we may suppose that we have also 1 In what follows WV shall omit stood that integers are to be ignored.
Landau, Hand&h der Lehre even der Verteil~unog der Primzahlen, p. 167. pp. 73 et seq. for the Q-functions : n 23 SOME PROBLEMS OF f)IOPHANTINE 227 APPROXIMATION of Genocchi and Schaar. Here p and q are integers of which one, is even and the other odd. By a suitable modification of Lindelijf’s argument, we establish the formula where B is an irrational number, which we may suppose to lie between - Z and 1, A is one of 2, 3, 4, X, a corresponding one of the same numbers, and 0 (1) stands for a function of n and 8 less in numericA value than an absolute constant.