## An asymptotic formula for the number of smooth values of a polynomial (1999)

Venue: | J. Number Theory |

Citations: | 11 - 1 self |

### BibTeX

@ARTICLE{Martin99anasymptotic,

author = {Greg Martin},

title = {An asymptotic formula for the number of smooth values of a polynomial},

journal = {J. Number Theory},

year = {1999},

pages = {448--460}

}

### OpenURL

### Abstract

Integers without large prime factors, dubbed smooth numbers, are by now firmly established as a useful and versatile tool in number theory. More than being simply a property of numbers that is conceptually dual to primality, smoothness has played a major role in the proofs of many results, from multiplicative questions to Waring’s problem to complexity

### Citations

155 | Sieve methods - Halberstam, Richert - 1974 |

134 |
Some problems of Partitio Numerorum III: on the expression of a number as a sum of primes
- Hardy, Littlewood
- 1923
(Show Context)
Citation Context ...e values simultaneously, where {L1, . . .,LK} is admissible if for every prime p, there exists an integer n such that none of L1(n), . . .,LK(n) is a multiple of p; subsequently, Hardy and Littlewood =-=[7]-=- proposed an asymptotic formula for how often this occurs. Schinzel and Sierpiński’s “Hypothesis H” [12] asserts that for an admissible set {F1, . . ., FK} of irreducible polynomials (integer-valued, ... |

72 | There are infinitely many Carmichael numbers
- Alford, Granville, et al.
- 1994
(Show Context)
Citation Context ...us on a special case for which a stronger theorem can be established. Shifted primes q + 1 without large prime factors played an important role in the recent proof by Alford, Granville, and Pomerance =-=[1]-=- that there are infinitely many Carmichael numbers; the counting function of these smooth shifted primes is precisely Φ(F; x, y) where F(t) = t + 1. More generally, for any nonzero integer a we define... |

59 |
A heuristic asymptotic formula concerning the distribution of prime numbers
- Bateman, Horn
- 1962
(Show Context)
Citation Context ...valued, naturally) of any degree, there are infinitely many integers n such that each of F1(n), . . .,FK(n) is prime; a quantitative version of this conjecture was first published by Bateman and Horn =-=[2]-=-. We must introduce some notation before we can describe the conjectured asymptotic formula, which we prefer to recast in terms of a single polynomial F rather than a set {F1, . . .,FK} of irreducible... |

55 |
Sur certaines hypothèses concernant les nombres premiers
- Schinzel, Sierpiński
- 1958
(Show Context)
Citation Context ...er n such that none of L1(n), . . .,LK(n) is a multiple of p; subsequently, Hardy and Littlewood [7] proposed an asymptotic formula for how often this occurs. Schinzel and Sierpiński’s “Hypothesis H” =-=[12]-=- asserts that for an admissible set {F1, . . ., FK} of irreducible polynomials (integer-valued, naturally) of any degree, there are infinitely many integers n such that each of F1(n), . . .,FK(n) is p... |

41 |
Integers without large prime factors
- Hildebrand, Tenenbaum
- 1993
(Show Context)
Citation Context ...ic progression, since Ψ(F; x, x 1/u ) = Ψ(qx + a, x 1/u ; q, a) − Ψ(a, x 1/u ; q, a) = Ψ(qx, x 1/u ; q, a) + O(1) = qxρ(u) ( ( 1 )) 1 + Oq,a q log x ( x ) = xρ(u) + OF,U log x (see the survey article =-=[8]-=-; here Ψ(x, y; q, a) denotes the number of y-smooth numbers up to x that are congruent to a (mod q)). This asymptotic formula is known unconditionally to hold for arbitrarily large values of U, which ... |

25 |
A new extension of dirichlet’s theorem on prime numbers
- Dickson
- 1904
(Show Context)
Citation Context ...howing another way in which these two properties are more than abstractly linked. There are conjectures about the distribution of prime values of polynomials that by now have become standard. Dickson =-=[4]-=- first conjectured that any K linear polynomials with integer coefficients forming an “admissible” set infinitely often take prime values simultaneously, where {L1, . . .,LK} is admissible if for ever... |

24 |
asymptotische verhalten von Summen über multiplikative Funktionen II, Acta
- Wirsing, Das
- 1967
(Show Context)
Citation Context ...ideas used in establishing the following proposition have been part of the “folklore” for some time, the literature does not seem to contain a result in precisely this form. Wirsing’s pioneering work =-=[14]-=-, for instance, requires g to be a nonnegative function and implies an asymptotic formula for Mg(x) without a quantitative error term; while Halberstam and Richert [6, Lemma 5.4] give an analogous res... |

11 |
Limitations to the equi–distribution of primes I
- Friedlander, Granville
- 1989
(Show Context)
Citation Context ...ounted are only polynomially large as a function of the number of terms y/q in the segment of the arithmetic progression. This restriction is not made merely for simplicity: Friedlander and Granville =-=[5]-=-, expanding on the ground-breaking ideas of Maier, showed that even in the case y = x, the asymptotic formula (B.1) can fail when the size of q is x/ log D x for arbitrarily large D. Certainly one can... |

9 |
On the number of solutions of polynomial congruences and Thue
- Stewart
- 1991
(Show Context)
Citation Context ...itely many of the θ(p) are zero. Huxley [9] gives a bound for σ that implies σ(f; p ν ) ≤ (deg f)p θ(p)/2 for any squarefree polynomial f and any prime power p ν (this estimate is improved by Stewart =-=[13]-=-, though it will suffice for our purposes as stated). In particular we see that σ(f; p ν ) ≤ (deg f)∆ 1/2 for all prime powers p ν , which establishes the lemma. Lemma 6.2. If f is a polynomial with i... |

8 |
A note on polynomial congruences, Recent Progress in Analytic Number Theory
- Huxley
- 1979
(Show Context)
Citation Context ... 1 uniformly for all prime powers p ν . Proof: Let ∆ be the discriminant of f, which is nonzero since f is squarefree, and write ∆ = ∏ p pθ(p) where all but finitely many of the θ(p) are zero. Huxley =-=[9]-=- gives a bound for σ that implies σ(f; p ν ) ≤ (deg f)p θ(p)/2 for any squarefree polynomial f and any prime power p ν (this estimate is improved by Stewart [13], though it will suffice for our purpos... |

8 |
Short sums of certain arithmetic functions
- Nair, Tenenbaum
- 1998
(Show Context)
Citation Context ... p | F(n) =⇒ p ≤ y}, the number of y-smooth values of F on arguments up to x; this generalizes the standard counting function Ψ(x, y) of y-smooth numbers up to x. Known upper bounds (see for instance =-=[11]-=-) and lower bounds (see [3]) for Ψ(F; x, y) do indicate the order of magnitude of Ψ(F; x, y) in certain ranges; however, in contrast to π(F; x), there seems to be no consensus concerning the expected ... |

6 |
Polynomial values free of large prime factors
- Dartyge, Martin, et al.
(Show Context)
Citation Context ...ber of y-smooth values of F on arguments up to x; this generalizes the standard counting function Ψ(x, y) of y-smooth numbers up to x. Known upper bounds (see for instance [11]) and lower bounds (see =-=[3]-=-) for Ψ(F; x, y) do indicate the order of magnitude of Ψ(F; x, y) in certain ranges; however, in contrast to π(F; x), there seems to be no consensus concerning the expected asymptotic formula for Ψ(F;... |

3 | Lower bounds for the number of smooth values of a polynomial, electronic preprint available online at http://xxx.lanl.gov/abs/math.NT/9807102 - Martin - 1998 |

2 |
Généralisation d’un théorème de Tchebycheff
- Nagel
- 1921
(Show Context)
Citation Context ...vant asymptotic formula is well-known. If f is a polynomial with integer coefficients with k distinct irreducible factors, then the values taken by σ on primes are k on average; more precisely, Nagel =-=[10]-=- showed that the asymptotic formula ∑ σ(f; p) logp = k log w + Of(1) (6.1) p p≤w holds for all w ≥ 2. For the purposes of establishing Proposition 2.9 (and, in the next section, Proposition 2.4), we n... |