An Asymptotic Formula for the Number of Smooth Values of a Polynomial
Greg Martin
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.
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