The distribution of integers with a divisor in a given interval

• 成果类型: Article
• 署名作者: Ford, Kevin
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.168.367
• 发表日期: 2008
• 页码: 367-433
• 关键词: probability; difference; primes

摘要

We determine the order of magnitude of H(x, y, z), the number of integers n <= 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 <= 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 <= x(1/2-epsilon). For every r >= 2, C > 1 and epsilon > 0, we determine the order of magnitude of H-r(x, y, z) uniformly for y large and y + y/(log y)(log4-1-epsilon) <= z <= min(y(C), x(1/2-epsilon)). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.