The distribution of integers with a divisor in a given interval

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Abstract

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 Hr(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 H1(x, y, z) for all x; y; z satisfying z ≤ x1/2-ε. For every r ≥ 2, C > 1 and " ε > 0, we determine the order of magnitude of Hr(x, y, z) uniformly for y large and y + y/(log y)log 4-1-ε ≤ z ≤ min(yC, x1/2-ε). As a consequence of these bounds, we settle a 1960 con-jecture of Erdos and some conjectures of Tenenbaum. One key element of the proofs is a new result on the distribution of uniform order statistics.

Original languageEnglish (US)
Pages (from-to)367-433
Number of pages67
JournalAnnals of Mathematics
Volume168
Issue number2
DOIs
StatePublished - 2008

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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