An Asymptotic Formula for the Number of Smooth Values of a Polynomial

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 analyses of factorization and primality-testing algorithms. In these last applications, what is needed is an understanding of the distribution of smooth numbers among the values taken by a polynomial, which is the subject of this paper. Specifically, we show a connection between the asymptotic number of prime values taken by a polynomial and the asymptotic number of smooth values so taken, showing 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 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, 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 polynomials. Let F (t) = F1(t) . . . FK(t) be the product of K distinct irreducible polynomials with integer coefficients. We say that the polynomial F is admissible if the set {F1, . . . , FK} is admissible, that is, if for every prime p there exists an integer n such that F (n) is not a multiple of p. Let π(F ; x) denote the number of positive integers n not exceeding x such that each Fi(n) is a prime (positive or negative). When F is an admissible polynomial, the size of π(F ; x) is heuristically C(F ) li(F ; x), these two quantities being defined as