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Arbitrarily Tight Bounds On The Distribution Of Smooth Integers
 Proceedings of the Millennial Conference on Number Theory
, 2002
"... This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve ..."
Abstract

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This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFTbased powerseries exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.
Smoothing "Smooth" Numbers
"... : An integer is called ysmooth if all of its prime factors are y. An important problem is to show that the ysmooth integers up to x are equidistributed amongst short intervals. In particular, for many applications we would like to know that if y is an arbitrarily small, fixed power of x then al ..."
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: An integer is called ysmooth if all of its prime factors are y. An important problem is to show that the ysmooth integers up to x are equidistributed amongst short intervals. In particular, for many applications we would like to know that if y is an arbitrarily small, fixed power of x then all intervals of length p x, up to x, contain, asymptotically, the same number of ysmooth integers. We come close to this objective by proving that such ysmooth integers are so equidistributed in intervals of length p xy 2+" , for any fixed " ? 0. * Partially supported by NSERC grant A5123 y An Alfred P. Sloan Research Fellow. Partially supported by the NSF. Smoothing "Smooth" Numbers John B. Friedlander and Andrew Granville Introduction. In this paper we will investigate the distribution, in short intervals, of integers without large prime factors . Good estimates, in such questions, have turned out to be surprisingly difficult. For instance, one might suppose that proving the ...