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Integers, without large prime factors, in arithmetic progressions, II
"... : We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan R ..."
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: We show that, for any fixed " ? 0, there are asymptotically the same number of integers up to x, that are composed only of primes y, in each arithmetic progression (mod q), provided that y q 1+" and log x=log q ! 1 as y ! 1: this improves on previous estimates. y An Alfred P. Sloan Research Fellow. Supported, in part, by the National Science Foundation Integers, without large prime factors, in arithmetic progressions, II Andrew Granville 1. Introduction. The study of the distribution of integers with only small prime factors arises naturally in many areas of number theory; for example, in the study of large gaps between prime numbers, of values of character sums, of Fermat's Last Theorem, of the multiplicative group of integers modulo m, of Sunit equations, of Waring's problem, and of primality testing and factoring algorithms. For over sixty years this subject has received quite a lot of attention from analytic number theorists and we have recently begun to attain a very pre...
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 ..."
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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.
Sieving and the ErdősKac Theorem
, 2006
"... We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. ..."
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We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature.
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 Hyp ..."
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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 ...
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"... Abstract. We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. Let ω(n) denote the number of distinct prime factors of the natural number n. ..."
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Abstract. We give a relatively easy proof of the ErdősKac theorem via computing moments. We show how this proof extends naturally in a sieve theory context, and how it leads to several related results in the literature. Let ω(n) denote the number of distinct prime factors of the natural number n. The average value of ω(n) as n ranges over the integers below x is