## Approximating the number of integers without large prime factors (2004)

Venue: | Mathematics of Computation |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Suzuki04approximatingthe,

author = {Koji Suzuki},

title = {Approximating the number of integers without large prime factors},

journal = {Mathematics of Computation},

year = {2004},

volume = {75},

pages = {1015--1024}

}

### OpenURL

### Abstract

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.

### Citations

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(Show Context)
Citation Context ...ers ≤ x and free of prime factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],=-=[4]-=-,[6],[8],[9],[10],[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime ... |

45 |
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Citation Context ...and free of prime factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],=-=[8]-=-,[9],[10],[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors ... |

30 |
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Citation Context ...free of prime factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],[8],=-=[9]-=-,[10],[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors exce... |

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Citation Context ...rs >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],[8],[9],[10],[11],[13],=-=[16]-=-). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors exceeding x1/u (0 <u) ap... |

27 |
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Citation Context ...s and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],[8],[9],[10],[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and =-=[15]-=-. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors exceeding x1/u (0 <u) approaches the value ρ(u)asx−→ ∞, where u =(logx)/ log y and ρ(u) is the uniq... |

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Citation Context ... free of prime factors >y. Estimates for Ψ(x, y) are very useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([2], [3], =-=[4]-=-, [5], [6], [7], [8], [10], [13]). We see good summaries for the investigations of Ψ(x, y) in [9] and [11]. Dickman [5] showed that the probability that a random integer between 1 and x has no prime f... |

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Citation Context ...small u. Hunter and Sorenson [13] gave some experimental data to show this fact. Dickman’s ρ can be computed by using the following equation: (1.1) ρ(u)= 1 � u ρ(t)dt for u ≥ 1. u u−1 Several authors =-=[2, 5, 14]-=- proposed other methods to efficiently calculate Dickman’s ρ. Hildebrand and Tenenbaum [11] gave an estimate of Ψ(x, y) which is accurate for large u. They showed that uniformly for 2 ≤ y ≤ x, � � � �... |

22 |
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Citation Context ...rime factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],[8],[9],[10],=-=[11]-=-,[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors exceeding x1/u... |

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Citation Context ...α ′ u | < min{0.0001, 1/(ū log x)}, whereαu is the unique solution to (1.2) and ū =min{u, y/ log y}. (3) Output h(x, y, α ′ u). Step (1) can be done by some sieve algorithms (for example, see [1] and =-=[12]-=-) using O(y/ log log y) operations. Step (2) requires bisection, and it can be performed using O(y(log log x)/ log y) operations [10]. Hence, the complexity of Algorithm HS is given by � � log log x (... |

9 | Prime sieves using binary quadratic forms
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Citation Context ...nd Patterson and Rumsey’s method (PR) to compute the dickman’s ρ in C++ programs. To find primes required for the algorithms HS, SO on RH, SU, and SU on RH, we used Atkin and Bernstein’s sieve method =-=[1]-=-, which uses O(y/log log y) operations and y1/2+o(1) bits of memory for finding all primes ≤ y. For Algorithm HS and SO on RH, we used Newton’s method for finding an estimate of αu. Instead of a value... |

9 | Approximating the number of integers free of large prime factors
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Citation Context ...factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],[8],[9],[10],[11],=-=[13]-=-,[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors exceeding x1/u (0 <... |

8 |
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Citation Context ...small u. Hunter and Sorenson [13] gave some experimental data to show this fact. Dickman’s ρ can be computed by using the following equation: (1.1) ρ(u)= 1 � u ρ(t)dt for u ≥ 1. u u−1 Several authors =-=[2, 5, 14]-=- proposed other methods to efficiently calculate Dickman’s ρ. Hildebrand and Tenenbaum [11] gave an estimate of Ψ(x, y) which is accurate for large u. They showed that uniformly for 2 ≤ y ≤ x, � � � �... |

7 |
Integers without large prime factors, Journal de Theorie des Nombres de Bordeaux 5
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(Show Context)
Citation Context ...algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],[8],[9],[10],[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in =-=[12]-=- and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors exceeding x1/u (0 <u) approaches the value ρ(u)asx−→ ∞, where u =(logx)/ log y and ρ(u) is... |

2 |
On the number of positive integers ≤x and free of prime factors >y
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(Show Context)
Citation Context ...≤ x and free of prime factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],=-=[6]-=-,[8],[9],[10],[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime fact... |

2 |
On the local behavior of Ψ(x, y
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(Show Context)
Citation Context ... of prime factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors ([3],[4],[6],[8],[9],=-=[10]-=-,[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no prime factors exceeding... |

2 |
A fast algorithm for approximately counting smooth numbers, ANTS-IV proceedings
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Citation Context ...on’s method can reduce the running time for finding the unique solution of (1.3), one can only prove quadratic convergence. Furthermore, assuming the validity of the Riemann Hypothesis (RH), Sorenson =-=[17]-=- proposed a modification of Hunter and Sorenson’s algorithm. The running time of this modified algorithm is roughly proportional to √ y. The author [18] gave another algorithm to evaluate (1.2). Let �... |

1 |
Bounding smooth integers, ANTS-III proceedings
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(Show Context)
Citation Context ...ntegers ≤ x and free of prime factors >y. Estimates for Ψ(x, y) are useful for many number-theoretic algorithms and modern cryptography. The behavior of Ψ(x, y) has been investigated by many authors (=-=[3]-=-,[4],[6],[8],[9],[10],[11],[13],[16]). We see good summaries for the investigations of Ψ(x, y) in [12] and [15]. Dickman [8] showed that the probability that a random integer between 1 and x has no pr... |