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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.
Lower bounds for the number of smooth values of a polynomial
, 1998
"... We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct ord ..."
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We investigate the problem of showing that the values of a given polynomial are smooth (i.e., have no large prime factors) a positive proportion of the time. Although some results exist that bound the number of smooth values of a polynomial from above, a corresponding lower bound of the correct order of magnitude has hitherto been established only in a few special cases. The purpose of this paper is to provide such a lower bound for an arbitrary polynomial. Various generalizations to subsets of the set of values taken by a polynomial are also obtained.
The number of cycles of specified normalized length in permutations
 In preparation
"... Abstract. We compute the limiting distribution, as n → ∞, of the number of cycles of length between γn and δn in a permutation of [n] chosen uniformly at random, for constants γ, δ such that 1/(k + 1) ≤ γ < δ ≤ 1/k for some integer k. This distribution is supported on {0, 1,..., k} and has 0th, ..."
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Abstract. We compute the limiting distribution, as n → ∞, of the number of cycles of length between γn and δn in a permutation of [n] chosen uniformly at random, for constants γ, δ such that 1/(k + 1) ≤ γ < δ ≤ 1/k for some integer k. This distribution is supported on {0, 1,..., k} and has 0th, 1st,..., kth moments equal. For more general choices of γ, δ we show that such a limiting distribution exists, which can be given explicitly in terms of certain integrals over intersections of hypercubes with halfspaces; these integrals are analytically intractable but a recurrence specifying them can to those of a Poisson distribution with parameter log δ γ be given. The results herein provide a basis of comparison for similar statistics on restricted classes of permutations. The distribution of the number of kcycles in a permutation of [n], for a fixed k, converges to a Poisson distribution with mean 1/k as k → ∞. In particular the mean number of kcycles and the variance of the number of kcycles are both 1/k whenever n ≥ k and n ≥ 2k respectively. If instead of holding k constant we let it vary with n, the number of αncycles in permutations of [n] approaches zero as n → ∞ with α fixed. So to investigate the number of cycles of long lengths, we must rescale and look at many cycle lengths at once. In particular, we consider the number of cycles with length in some interval [γn, δn] as n → ∞. The expectation of the number of cycles with length in this 1/k, which approaches the constant log δ/γ as n grows large. By analogy with the fixedk case we might expect the number of cycles with length in this interval to be Poissondistributed. But this cannot be the case, because there is room for at most 1/γ cycles of length at least γn, and the Poisson distribution can take arbitrarily large values. In the case where 1/γ and 1/δ lie in the same interval [1/(k + 1), 1/k] for some integer k, the limit distribution has the same first k moments as Poisson(log δ/γ). For general γ and δ the situation is considerably more complex but a limit distribution still exists. interval is ∑ δn k=γn
Divisibility, Smoothness and Cryptographic Applications
, 2008
"... This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in var ..."
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This paper deals with products of moderatesize primes, familiarly known as smooth numbers. Smooth numbers play an crucial role in information theory, signal processing and cryptography. We present various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences. We then turn our attention to cryptographic applications in which smooth numbers play a pivotal role. 1 1