## Arbitrarily Tight Bounds On The Distribution Of Smooth Integers (2002)

Venue: | Proceedings of the Millennial Conference on Number Theory |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Bernstein02arbitrarilytight,

author = {Daniel J. Bernstein},

title = {Arbitrarily Tight Bounds On The Distribution Of Smooth Integers},

booktitle = {Proceedings of the Millennial Conference on Number Theory},

year = {2002},

pages = {49--66}

}

### OpenURL

### Abstract

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 FFT-based power-series exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.

### Citations

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Citation Context ... , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31], [32], [51], [18=-=-=-], [59], and [57]. De Bruijn in [25] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further information. I am not aware of any attempts... |

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Citation Context ...of to high accuracy; see [95], [20], [71, section 9], [50], [49], [77], [21, section 3], [70], and [3, section 4]. For asymptotics as u ! 1 see [15], [26], [17], [64], [88], and [96]. Hildebrand in [61] showed that the error j (H; y)=H(u) 1j, where H = y u , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior resul... |

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Citation Context ... ! Q such that every nonempty subset of fr 2 R : f r 6= 0g has a least element|is widely known but is not equipped with a useful notion of distribution. This larger ring was introduced by Malcev; see =-=-=-[86] for more information. 3. Bounds on the distribution of smooth integers Fix positive integers y and . For each prime p y select a real number p p, preferably as small as possible, with lg p 2 Z... |

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Citation Context ... not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31], [32], [51], [18], [59], and [57]. De Bruijn in [25=-=-=-] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further information. I am not aware of any attempts to compute this approximation. Ran... |

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Citation Context ... Figure 2. How to compute lower bounds ons(2 0 ; y),s(2 1= ; y), . . . ,s(2 h 1= ; y). A split-radix FFT uses (12+o(1))h lg h additions and multiplications in R to multiply in R[x 1= ]=x h ; see [9]. Brent's exponentiation algorithm in [11] then uses (88 + o(1))h lg h additions and multiplications in R to compute g mod x h given log g mod x h . The constant 88 can be improved to 34; see [10]. ... |

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Citation Context ... H in the geometric progression 2 0 ; 2 1= ; : : : ; 2 h 2= ; 2 h 1= . See Figure 2. g h 7!s(2 h ; y) distr g Figure 1. For y = 7 and = 5: Graphs of g, distr g, and h 7!s(2 h ; y), restricted to [0; =-=-=-10]. Vertical range [0; 143]. 6 DANIEL J. BERNSTEIN Enumerate primes p y 2; 3; 5; : : : ## For each p,snd multiple of 1= no smaller than lg p lg 2; lg 3; lg 5; : : : ## Compute P py x lg p + 1 2 x 2 ... |

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Citation Context ...d out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further information. I am not aware of any attempts to compute this approximation. Rankin in [85] observed thats(H; y) H s = Q py (1 p s ) for any s > 0. This upper bound is minimized when s satises P py (log p)=(p s 1) = log H. Hildebrand and Tenenbaum in [65] showed that the approximation ... |

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Citation Context ...Foundation under grants DMS{ 9600083 and DMS{9970409, and by the Mathematical Sciences Research Institute. 1 2 DANIEL J. BERNSTEIN Other work. There are many limited-precision approximations tos. See =-=[8-=-2], [72], [66], and [81] for detailed surveys of the results and the underlying techniques. Dickman in [29] observed that lim y!1s(y u ; y)=y u = (u) for u > 0. Here is the unique continuous function... |

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Citation Context ... this approximation. Rankin in [85] observed thats(H; y) H s = Q py (1 p s ) for any s > 0. This upper bound is minimized when s satises P py (log p)=(p s 1) = log H. Hildebrand and Tenenbaum in [65] showed that the approximation 1 s 2 X py p s (log p) 2 (p s 1) 2 1=2 H s Y py 1 1 p s tos(H; y), with the same choice of s as in Rankin's bound, has error at most a constant (again not compute... |

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Citation Context ...product of many ratios of the forms(H; y)= (H 1=(1+) ; y), for varying H, must be large. Quite a few of the ratios have to be at least about H =(1+) . For uniform lower bounds see [41], [4], [52], [75], [69], and [98]. See [66] and [40] for precise asymptotics when is not very small and log y is noticeably bigger than (log H) 5=6 . How depends on . Dene as the maximum of (lg p)= lg p 1 for ... |

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Citation Context ... +335653893002534131235548574x 99 + . The bounds in this paper boil down to a known algorithm to compute the exact coecients of this series. For asymptotic estimates see [19], [76], [6], and [83]. Coprime pairs. Consider the series X n1 ;n2 [n 1 is y-smooth][n 2 is y-smooth][gcd fn 1 ; n 2 g = 1] x lg n1 1 x lg n2 2 in two variables x 1 ; x 2 . This series is the product over smooth primes p ... |

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Citation Context ... bound ons(H; y; i) for each i and various H. One can save time by working in the smaller ring Q[(Z=m) ] and ignoring primes that divide m. For previous work see [16], [36], [53], [54], [38], [39], [=-=33]-=-, [5], [47], [48], [93], [97], and [34]. See [43] for more information on monoid rings and group rings. Numberselds. Let K be a numberseld, R its ring of integers. A nonzero ideal n of R is y-smooth i... |

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Citation Context ...Consequently the product of many ratios of the forms(H; y)= (H 1=(1+) ; y), for varying H, must be large. Quite a few of the ratios have to be at least about H =(1+) . For uniform lower bounds see [41], [4], [52], [75], [69], and [98]. See [66] and [40] for precise asymptotics when is not very small and log y is noticeably bigger than (log H) 5=6 . How depends on . Dene as the maximum of (l... |

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Citation Context ...H; y; i) for each i and various H. One can save time by working in the smaller ring Q[(Z=m) ] and ignoring primes that divide m. For previous work see [16], [36], [53], [54], [38], [39], [33], [5], [=-=47]-=-, [48], [93], [97], and [34]. See [43] for more information on monoid rings and group rings. Numberselds. Let K be a numberseld, R its ring of integers. A nonzero ideal n of R is y-smooth if it has no... |

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Citation Context ... ons(H; y; i) for each i and various H. One can save time by working in the smaller ring Q[(Z=m) ] and ignoring primes that divide m. For previous work see [16], [36], [53], [54], [38], [39], [33], [=-=5]-=-, [47], [48], [93], [97], and [34]. See [43] for more information on monoid rings and group rings. Numberselds. Let K be a numberseld, R its ring of integers. A nonzero ideal n of R is y-smooth if it ... |

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Citation Context ... (log p) 2 (p s 1) 2 1=2 H s Y py 1 1 p s tos(H; y), with the same choice of s as in Rankin's bound, has error at most a constant (again not computed) times 1=u + (log y)=y. Hunter and Sorenson in [6=-=-=-7] showed that one can compute these approximations in time roughly y. Sorenson subsequently suggested replacing each P py with P py c + P y csfor some c between 0 and 1, then approximating P y csby a... |

10 |
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Citation Context ... 1+2x+4x 2 + +335653893002534131235548574x 99 + . The bounds in this paper boil down to a known algorithm to compute the exact coecients of this series. For asymptotic estimates see [19], [76], [6], and [83]. Coprime pairs. Consider the series X n1 ;n2 [n 1 is y-smooth][n 2 is y-smooth][gcd fn 1 ; n 2 g = 1] x lg n1 1 x lg n2 2 in two variables x 1 ; x 2 . This series is the product over s... |

9 |
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Citation Context ... 1=. For previous work see [68] (in the case K = Q[ p 1]), [42], [35], [55], [72], [73], [79], and [12]. In some applications|notably integer factorization with the numberseld sieve, as described in [=-=74]-=-|one wants to know the distribution of smooth elements of R. A fractional-power-series exponentiation over Q[G], where G is the ideal class group of R, produces bounds on the distribution of smooth id... |

7 |
On the distribution of integers having no large prime factors
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Citation Context ...uently the product of many ratios of the forms(H; y)= (H 1=(1+) ; y), for varying H, must be large. Quite a few of the ratios have to be at least about H =(1+) . For uniform lower bounds see [41], [4], [52], [75], [69], and [98]. See [66] and [40] for precise asymptotics when is not very small and log y is noticeably bigger than (log H) 5=6 . How depends on . Dene as the maximum of (lg p)=... |

7 | editors. Algorithmic Number Theory - Buhler, Stevenhagen - 2008 |

7 | A differential delay equation arising from the sieve - Cheer, Goldston |

7 |
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Citation Context ...nder grants DMS{ 9600083 and DMS{9970409, and by the Mathematical Sciences Research Institute. 1 2 DANIEL J. BERNSTEIN Other work. There are many limited-precision approximations tos. See [82], [72], =-=[6-=-6], and [81] for detailed surveys of the results and the underlying techniques. Dickman in [29] observed that lim y!1s(y u ; y)=y u = (u) for u > 0. Here is the unique continuous function satisfying ... |

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Citation Context ...1. One can rapidly compute and some useful variants of to high accuracy; see [95], [20], [71, section 9], [50], [49], [77], [21, section 3], [70], and [3, section 4]. For asymptotics as u ! 1 see [1=-=-=-5], [26], [17], [64], [88], and [96]. Hildebrand in [61] showed that the error j (H; y)=H(u) 1j, where H = y u , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, ... |

6 |
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Citation Context ...the error j (H; y)=H(u) 1j, where H = y u , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31], [32], [51], [18], [59], and [57]. De Bruijn in [25] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further i... |

6 |
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Citation Context ... [24] in 2000 as I was preparing the bibliography for this paper. 5. Generalizations and variants Omitting tiny primes. One can replace fp yg by a subset, such as fp : zs yg. For previous work see [37], [89], and [90]. Squarefree integers. One can restrict the powers of p that are allowed to appear: for example, one can replace 1 + x lg p + x 2 lg p + by 1 + x lg p to bound the distribution o... |

6 |
intervals containing numbers without large prime factors, ibid
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Citation Context ...y the product of many ratios of the forms(H; y)= (H 1=(1+) ; y), for varying H, must be large. Quite a few of the ratios have to be at least about H =(1+) . For uniform lower bounds see [41], [4], [52], [75], [69], and [98]. See [66] and [40] for precise asymptotics when is not very small and log y is noticeably bigger than (log H) 5=6 . How depends on . Dene as the maximum of (lg p)= lg p ... |

6 |
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Citation Context ...at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31], [32], [51], [18], [59=-=-=-], and [57]. De Bruijn in [25] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further information. I am not aware of any attempts to co... |

6 |
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Citation Context ...primes p of 1 + x lg p w + x 2 lg p w 2 + . The exponentiation here is faster than in the case of coprime pairs, because the exponents of w are very small. For previous work see [28], [56], and [63=-=]-=-. Semismoothness. The analysis and optimization of factoring algorithms often relies on the distribution of positive integers n that have no prime divisors larger than z and at most one prime divisor ... |

6 |
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Citation Context ... each i and various H. One can save time by working in the smaller ring Q[(Z=m) ] and ignoring primes that divide m. For previous work see [16], [36], [53], [54], [38], [39], [33], [5], [47], [48], [=-=93]-=-, [97], and [34]. See [43] for more information on monoid rings and group rings. Numberselds. Let K be a numberseld, R its ring of integers. A nonzero ideal n of R is y-smooth if it has no prime divis... |

5 |
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Citation Context .... 1 2 DANIEL J. BERNSTEIN Other work. There are many limited-precision approximations tos. See [82], [72], [66], and [81] for detailed surveys of the results and the underlying techniques. Dickman in [29] observed that lim y!1s(y u ; y)=y u = (u) for u > 0. Here is the unique continuous function satisfying (u) = 1 for 0s 1 and u(u) = R u u 1 (t) dt for u > 1. One can rapidly compute and some ... |

5 |
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Citation Context ...lg pe=, and multiplied the resulting series; I replace k lg p by kd lg pe=, which is not quite as small but is better suited for exponentiation. For limited-precision estimates see [44], [45], and [46]. Number of prime factors. The series P n [n is y-smooth] x lg n wsn) in two variables x; w, where n) = P p ord p n, is the product over smooth primes p of 1 + x lg p w + x 2 lg p w 2 + . The ex... |

4 |
Multiple-precision zero- methods and the complexity of elementary function evaluation
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Citation Context ...s(2 0 ; y),s(2 1= ; y), . . . ,s(2 h 1= ; y). A split-radix FFT uses (12+o(1))h lg h additions and multiplications in R to multiply in R[x 1= ]=x h ; see [9]. Brent's exponentiation algorithm in [11] then uses (88 + o(1))h lg h additions and multiplications in R to compute g mod x h given log g mod x h . The constant 88 can be improved to 34; see [10]. One can enumerate primes p y as describe... |

4 |
Répartition statistique des entiers sans grand facteur premier dans les progressions arithmétiques
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Citation Context ...ous H. One can save time by working in the smaller ring Q[(Z=m) ] and ignoring primes that divide m. For previous work see [16], [36], [53], [54], [38], [39], [33], [5], [47], [48], [93], [97], and [=-=34]-=-. See [43] for more information on monoid rings and group rings. Numberselds. Let K be a numberseld, R its ring of integers. A nonzero ideal n of R is y-smooth if it has no prime divisors of norm larg... |

4 |
Integers without large prime factors
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- 1981
(Show Context)
Citation Context ....e., a lower bound ons(H; y; i) for each i and various H. One can save time by working in the smaller ring Q[(Z=m) ] and ignoring primes that divide m. For previous work see [16], [36], [53], [54], [=-=38]-=-, [39], [33], [5], [47], [48], [93], [97], and [34]. See [43] for more information on monoid rings and group rings. Numberselds. Let K be a numberseld, R its ring of integers. A nonzero ideal n of R i... |

4 |
The number of positive integers free of prime divisors >x c ,anda problem of S.S
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(Show Context)
Citation Context ...ror j (H; y)=H(u) 1j, where H = y u , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31], [32], [51], [18], [59], and [57]. De Bruijn in [25] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further informa... |

3 |
Vijayaraghavan: ‘On the largest prime divisors of numbers
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(Show Context)
Citation Context ... that the error j (H; y)=H(u) 1j, where H = y u , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31], [32], [51], [18], [59], and [57]. De Bruijn in [25] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for fur... |

3 | On the number of positive integers � x and free of prime factors Simon Stevin 40
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(Show Context)
Citation Context ... 1j, where H = y u , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31=-=-=-], [32], [51], [18], [59], and [57]. De Bruijn in [25] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further information. I am not awa... |

3 |
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(Show Context)
Citation Context ... = y u , is at most a constant (which has not been computed) times (log(u + 1))= log y if u 1, H 3, and log y (log log H) 1:667 . For prior results see [23], [16], [84], [22], [27], [31], [32], [51=-=-=-], [18], [59], and [57]. De Bruijn in [25] pointed out that H R H 0 (u (log t)= log y) d(btc=t) is a better approximation tos(H; y). See [87] and [66] for further information. I am not aware of any at... |

3 |
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(Show Context)
Citation Context ...g H) s(H; y)= (H 1=(1+) ; y). The point is that one already has a good estimate for the ratios(H; y)= (H 1=(1+) ; y), namely 1 + log H. Here is a brief summary of the literature: Hildebrand in [60] proved that, for an extremely broad range of H and y, the ratio is at most about 1 + (H= (H 1=(1+) ; y)) log y. Hildebrand in [62] proved that, when is not very small, the ratio is at most H =... |

3 | Analytic computational complexity - Traub - 1976 |

2 |
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(Show Context)
Citation Context ...l number of relevant terms of g 1 and g 2 , hence the total time needed, can be quite a bit smaller thans(H; y). Notes. The ideas in this paper evolved as follows. I presented the exactsalgorithms in =-=[-=-7]. That paper was not phrased in the language of series; I used logarithms and merely because additions are faster than multiplications. I subsequently noticed that reducing would produce bounds on... |

2 |
Bounding smooth integers (extended abstract), in [14
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- 1998
(Show Context)
Citation Context ... bounds onsat high speed. In 1997, I rephrased the algorithms in the language of series, and realized the relevance of fast power-series exponentiation. An extended abstract of this paper appeared in =-=[8-=-]. I found Coppersmith's article [24] in 2000 as I was preparing the bibliography for this paper. 5. Generalizations and variants Omitting tiny primes. One can replace fp yg by a subset, such as fp :... |

2 | Computational perspectives on number theory - Buell, Teitelbaum - 1998 |

2 |
A probabilistic approach to a dierential-dierence equation arising in analytic number theory
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Citation Context ... > 0. Here is the unique continuous function satisfying (u) = 1 for 0s 1 and u(u) = R u u 1 (t) dt for u > 1. One can rapidly compute and some useful variants of to high accuracy; see [95], [20], [71, section 9], [50], [49], [77], [21, section 3], [70], and [3, section 4]. For asymptotics as u ! 1 see [15], [26], [17], [64], [88], and [96]. Hildebrand in [61] showed that the error j (H; y)=H... |

2 |
Fermat’s Last Theorem (Case 1) and the Wieferich criterion
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(Show Context)
Citation Context ... these bounds interesting is that they can be computed at extremely high speed, even when y is large. See section 3. As far as I know, thesrst publication of bounds of this type was by Coppersmith in =-=[24]-=-. Coppersmith showed how to compute an arbitrarily tight lower bound on a variant ofsin a reasonable amount of time. The main improvements in this paper are the fast algorithms in section 3 and the al... |

2 |
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(Show Context)
Citation Context ...Sorenson subsequently suggested replacing each P py with P py c + P y csfor some c between 0 and 1, then approximating P y csby an integral; this saves time at the expense of accuracy. See [92] and [30] for more information ons(H; y) when y is extremely small: in particular, on the accuracy of approximations such ass(H; 5) (log H) 3 =6(log 2)(log 3)(log 5). Notation. lg means log 2 . [ ] means... |

2 |
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(Show Context)
Citation Context ...g norm p + over smooth prime ideals p. One obtains a lower bound on distr f by increasing each lg norm p to a nearby multiple of 1=. For previous work see [68] (in the case K = Q[ p 1]), [42], [35]=-=-=-, [55], [72], [73], [79], and [12]. In some applications|notably integer factorization with the numberseld sieve, as described in [74]|one wants to know the distribution of smooth elements of R. A fra... |