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**Extra resources for Collected papers of G.H. Hardy; including joint papers with J.E. Littlewood and others**

**Example text**

Akademie len, vol. 53, pv 289 ; London, Math. Annah, i6id. der Wiss. xu Ntinchen, WI. 27, p* 101, and Math. , p. 322. 21 SOME PMOl3LEMS OF DIOPHANTINE APPROXIMATION 226 Applying the general principle once more we deduce a sequence of values which (ne) + 0, (n2S) + 0 simultaneously. P. An analogous method may be applied in the case of the general power 18~. As in the course of this proof we obtain a sequence for which simultaneously, we thus prove the theorem when u = 0 for the general polynomial f(n).

Iff(n) = 3n we can obtain an excluded interval whose length is as near Q as we please, and so on, while if f(n) = n! it’ is (as is well known) possible to choose B so that (n! a) hax a unique limit. Thus (n ! e) * 0. 2. The first object of this investigation theorem :- Theorem seqwnce cm 20 1912, 4 (with natiml Cmgress 1 R {f f(n) be founii fw J. E. LittIewood) of ikfathematicians, has been to prove the following is a polynomial in n, zuith integral coeficients, thert a which (,fr8) + a. Proceeding8 Cambridge, of the 6th 1912, i.

HARDY AND J. E, LITTLEWOOD If p tends to a rational place, it is known that f(q) tends to a limit or becomes definitely infinite of order +. By arguments depending upon the formulae of transformation of the S-functions, and similar in principle to, though simpler than, those of 5 4, we prove Theorem 7 Wh l q tends to any irrational No better result thm this is true irt yenerd. irratimzals deJCined in Theorem 5, theyz. place on the circle of convergeme, If q + @, where 9 is one of the m=o {(l -Iqj)-f].