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56
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
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Improved CRT algorithm for class polynomials in genus 2.” In: Algorithmic Number Theory — ANTSX. Edited by Everett Howe and Kiran Kedlaya
 Mathematical Science Publishers
"... Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring whi ..."
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Abstract. We present a generalization to genus 2 of the probabilistic algorithm in Sutherland [28] for computing Hilbert class polynomials. The improvement over the algorithm presented in [5] for the genus 2 case, is that we do not need to find a curve in the isogeny class with endomorphism ring which is the maximal order: rather we present a probabilistic algorithm for “going up ” to a maximal curve (a curve with maximal endomorphism ring), once we find any curve in the right isogeny class. Then we use the structure of the Shimura class group and the computation of (ℓ, ℓ)isogenies to compute all isogenous maximal curves from an initial one. This article is an extended version of the version published at ANTS X. 1.
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."
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Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents
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 <y ≤ x,whereɛ is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, ..."
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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.
On Quadratic Polynomials for the Number Field Sieve
 Australian Computer Science Communications
, 1997
"... . The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this pa ..."
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. The newest, and asymptotically the fastest known integer factorisation algorithm is the number field sieve. The area in which the number field sieve has the greatest capacity for improvement is polynomial selection. The best known polynomial selection method finds quadratic polynomials. In this paper we examine the smoothness properties of integer values taken by these polynomials. Given a quadratic NFS polynomial f , let \Delta be its discriminant. We show that a prime p can divide values taken by f only if (\Delta=p) = 1. We measure the effect of this residuosity property on the smoothness of fvalues by adapting a parameter ff, developed for analysis of MPQS, to quadratic NFS polynomials. We estimate the yield of smooth values for these polynomials as a function of ff, and conclude that practical changes in ff might bring significant changes in the yield of smooth and almost smooth polynomial values. Keywords: integer factorisation, number field sieve 1
The twoparameter PoissonDirichlet point process
, 2007
"... The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtai ..."
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The twoparameter PoissonDirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Relying on this, we apply the theory of point processes to reveal mathematical structure of the twoparameter PoissonDirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we will be able to extend several results previously known for the oneparameter case, and the MarkovKrein identity for the generalized Dirichlet process is discussed from a point of view of functional analysis based on the twoparameter PoissonDirichlet distribution. 1
On a combinatorial method for counting smooth numbers in sets of integers
 J. Number Theory
"... In this paper we prove a result for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth ” sums a + b, a ∈ A, b ∈ B. The proof of this r ..."
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In this paper we prove a result for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth ” sums a + b, a ∈ A, b ∈ B. The proof of this result is completely combinatorial and elementary. 1
On the largest prime factors of n and n+1
, 1978
"... If n >= 2 is an integer, let p(n) denote the largest prime factor of IZ. For every x>O and every t,O~ttl, let A(x,t) denote the number of nix with P(n)rx’. A wellknown result due to Dickman [4] and others is ..."
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If n >= 2 is an integer, let p(n) denote the largest prime factor of IZ. For every x>O and every t,O~ttl, let A(x,t) denote the number of nix with P(n)rx’. A wellknown result due to Dickman [4] and others is
On the convex closure of the graph of modular inversions
, 2006
"... In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set Gn = {(a,b) : a,b ∈ Z,ab ≡ 1 (mod n), 1 ≤ a,b ≤ n − 1}. The heuristic is based on an asymptotic formula of Rényi and Sulanke. After describing two algorithms to det ..."
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In this paper we give upper and lower bounds as well as a heuristic estimate on the number of vertices of the convex closure of the set Gn = {(a,b) : a,b ∈ Z,ab ≡ 1 (mod n), 1 ≤ a,b ≤ n − 1}. The heuristic is based on an asymptotic formula of Rényi and Sulanke. After describing two algorithms to determine the convex closure, we 1 compare the numeric results with the heuristic estimate. The numeric results do not agree with the heuristic estimate — there are some interesting peculiarities for which we provide a heuristic explanation. We then describe some numerical work on the convex closure of the graph of random quadratic and cubic polynomials over Zn. In this case the numeric results are in much closer agreement with the heuristic, which strongly suggests that the the curve xy = 1 (mod n) is “atypical”. 1
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