On the number of positive integers . . .
نویسندگان
چکیده
1. Introduction Let P(x,y) denote the number of integers specified in the title. A number of estimates and asymptotic formulae for this function have been given (cf. [1] and the literature mentioned there). Recently DE BRUIJN ([2]) proved an asymptotic formula for log f(x,y) which holds uniformly for 2 < y < x. Part of the proof consisted of showing that P(x y) > C 71(y) + u\ where u = [(log x)í(to u g y)] • It is the purpose of this note to extend this inequality to an asymptotic formula (which isweaker than DE BRUIJN's result). In fact we shall prove : Theorem 1 : For 2 < y < x we have for x-+ oo, uniformly in y, log P (x,y) log 7E (y) + u a (1) where u = [(log x)/(log y)]. We remark that this of course follows from DE BRUIJN's theorem. Our interest lies mainly in giving a short fairly straightforward proof. For some ranges of values of y (1) is nearly trivial and the most interesting part of the proof concerns the range (log x)` < y < (log v)' +F 7 3
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تاریخ انتشار 1966