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The distribution of integers with at least two divisors in a short interval

Kevin Ford·Gerald Tenenbaum
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Abstract

Let H(x,y,z) be the number of integers $\le x$ with a divisor in (y,z] and let H_1(x,y,z) be the number of integers $\le x$ with exactly one such divisor. When y and z are close, it is expected that H_1(x,y,z) H(x,y,z), that is, an integer with a divisor in (y,z] usually has just one. We determine necessary and sufficient conditions on y and z so that H_1(x,y,z) H(x,y,z). In doing so, we answer an open question from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223.

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Kevin Ford, Gerald Tenenbaum. 2007-02-09. The distribution of integers with at least two divisors in a short interval. https://arxiv.org/abs/math/0607460

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