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

Abstract

Although we expect to find many smooth numbers (i.e., numbers with no large prime factors) among the values taken by a polynomial with integer coefficients, it is unclear what the asymptotic number of such smooth values should be; this is in contrast to the related problem of counting the number of prime values of a polynomial, for which Bateman and Horn published a conjectured asymptotic formula that is widely believed to be true. We discuss how to employ the Bateman-Horn conjecture to derive an asymptotic formula for the number of smooth values of a polynomial, with the smoothness parameter in a non-trivial range. This conditional result provides a believable heuristic for the number of smooth integers among all values {F(n)}, and also among the values {F(p)} on prime arguments only.