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24
Limit Theorems for the Number of Summands in Integer Partitions
, 2000
"... Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramertype large deviations and are proved by Mellin transform and th ..."
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Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramertype large deviations and are proved by Mellin transform and the twodimensional saddlepoint method. Applications of these results include partitions into positive integers, into powers of integers, into integers [j ], # > 1, into aj + b, etc.
Approximating the number of integers free of large prime factors
 Math. Comp
, 1997
"... Abstract. Define Ψ(x, y) to be the number of positive integers n ≤ x such that n has no prime divisor larger than y. We present a simple algorithm that log log x approximates Ψ(x, y) inO(y { log y + 1}) floating point operations. log log y This algorithm is based directly on a theorem of Hildebrand ..."
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Abstract. Define Ψ(x, y) to be the number of positive integers n ≤ x such that n has no prime divisor larger than y. We present a simple algorithm that log log x approximates Ψ(x, y) inO(y { log y + 1}) floating point operations. log log y This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other known approximations, including the wellknown approximation Ψ(x, y) ≈ xρ(log x / log y), where ρ(u) is Dickman’s function. 1.
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
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 Research Fe ..."
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Cited by 8 (1 self)
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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...
Subtleties in the distribution of the numbers of points on elliptic curves over a finite prime field
 Journal of the London Mathematical Society
, 1999
"... Three questions concerning the distribution of the numbers of points on elliptic curves over a finite prime field are considered. First, the previously published bounds for the distribution are tightened slightly. Within these bounds, there are wild fluctuations in the distribution, and some heurist ..."
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Three questions concerning the distribution of the numbers of points on elliptic curves over a finite prime field are considered. First, the previously published bounds for the distribution are tightened slightly. Within these bounds, there are wild fluctuations in the distribution, and some heuristics are discussed (supported by numerical evidence) which suggest that numbers of points with no large prime divisors are unusually prevalent. Finally, allowing the prime field to vary while fixing the field of fractions of the endomorphism ring of the curve, the order of magnitude of the average order of the number of divisors of the number of points is determined, subject to assumptions about primes in quadratic progressions. There are implications for factoring integers by Lenstra’s elliptic curve method. The heuristics suggest that (i) the subtleties in the distribution actually favour the elliptic curve method, and (ii) this gain is transient, dying away as the factors to be found tend to infinity. 1.
Reciprocals of certain large additive functions
 225—231; MR0619450 (82k:10053). involving Arithmetric Functions 12
, 1981
"... 1. Introduction and statement of results ..."
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.
Approximating the number of integers without large prime factors
 Mathematics of Computation
, 2004
"... Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ
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Abstract. Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors>y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x) 1+ɛ <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O ( � (log x)(log y)). We give numerical results which show that this algorithm provides accurate estimates for Ψ(x, y) andisfaster than conventional methods such as algorithms exploiting Dickman’s function. 1.
Fast Bounds on the Distribution of Smooth Numbers
, 2006
"... Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our fi ..."
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Let P(n) denote the largest prime divisor of n, andlet Ψ(x,y) be the number of integers n ≤ x with P(n) ≤ y. Inthispaper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y 2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if log y is a fractional power of log x, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in log y, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.