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The distribution of integers with a divisor in a given interval | | Kevin Ford
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18 Dec 2003 | | Subject: | Number Theory MSC-class: 11N25 (Primary) 62G30 (Secondary) | math.NT | | Abstract: | We determine the order of magnitude of H(x,y,z), the number of integers nle 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 nle 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 zle x^{0.49}. For every rge 2, 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 - epsilon} le z le min(y^{10r},x^{0.49}). 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. | | Source: | arXiv, math.NT/0401223 | | Services: | Forum | Review | PDF | Favorites | |
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