1999/09/30 by Greg Martin · 1 citation
Mathematics · #math.NT #msc:11N32 #msc:11N25
published as J. Number Theory 93 (2002), no. 2, 108-182. · 57 pages. Revised version - an appendix has been added and some other material rewritten slightly
arxiv created 2000/08/22 · arxiv updated 2009/11/30
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.