The distribution of integers with a divisor in a given interval
| dc.creator | Ford, Kevin | |
| dc.date | 2004-01-18 | |
| dc.date | 2008-11-06 | |
| dc.date.accessioned | 2026-07-07T10:15:45Z | |
| dc.date.available | 2026-07-07T10:15:45Z | |
| dc.description | We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, $C>1$ and $ε>0$, we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - ε} \le z \le \min(y^{C},x^{1/2-ε}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key element of the proofs is a new result on the distribution of uniform order statistics. | |
| dc.description | Final version. Greatly simplified proof of Lemma 4.7 in Sec. 10, references updated, other minor corrections | |
| dc.identifier | https://arxiv.org/abs/math/0401223 | |
| dc.identifier | http://arxiv.org/abs/math/0401223 | |
| dc.identifier | Annals of Math. (2) 168 (2008), 367-433 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/173278 | |
| dc.subject | Number Theory | |
| dc.subject | 11N25 (Primary) 62G30 (Secondary) | |
| dc.title | The distribution of integers with a divisor in a given interval | |
| dc.type | text |