An Asymptotic Formula for the Number of Smooth Values of a Polynomial
| dc.creator | Martin, Greg | |
| dc.date | 1999-09-29 | |
| dc.date | 2000-08-22 | |
| dc.date.accessioned | 2026-07-07T05:30:57Z | |
| dc.date.available | 2026-07-07T05:30:57Z | |
| dc.description | 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. | |
| dc.description | 57 pages. Revised version - an appendix has been added and some other material rewritten slightly | |
| dc.identifier | https://arxiv.org/abs/math/9909180 | |
| dc.identifier | http://arxiv.org/abs/math/9909180 | |
| dc.identifier | J. Number Theory 93 (2002), no. 2, 108-182. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/79173 | |
| dc.subject | Number Theory | |
| dc.subject | 11N32, 11N25 | |
| dc.title | An Asymptotic Formula for the Number of Smooth Values of a Polynomial | |
| dc.type | text |