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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.
On Positive Integers ≤x with Prime Factors ≤t log x
"... . It is not difficult to estimate the function /(x; y), which counts integers x, free of prime factors ? y, by "smooth" functions whenever y log 1=2 x or y is a fixed power of x. This can be extended to y ! log 3=4 x, and y ? log 2+" x under the assumption of the Riemann Hypothesis. The re ..."
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. It is not difficult to estimate the function /(x; y), which counts integers x, free of prime factors ? y, by "smooth" functions whenever y log 1=2 x or y is a fixed power of x. This can be extended to y ! log 3=4 x, and y ? log 2+" x under the assumption of the Riemann Hypothesis. The real difficulty lies when y is a fixed multiple of log x and, in this paper, we investigate the set of integers x, free of prime factors ? t log x, by estimating various functions related to /(x; t log x). 1. INTRODUCTION. Define S(x; y) to be the set of positive integers x, composed only of prime factors y. The cardinality of this set, /(x; y), is called the DickmanDe Bruijn function and has been extensively investigated by many authors (see [14] for a review). In this section we will give some wellknown results about /(x; y) and sketch proofs of smooth asymptotic estimates when y ! log 1=2 x and when y is a fixed power of x. We also indicate how, in the literature, these have been ...