THE DISTRIBUTION OF INTEGERS WITH AT LEAST TWO DIVISORS IN A SHORT INTERVAL

DOI10.1093/QMATH/HAM001zbMATH Open1213.11170arXivmath/0607460OpenAlexW2075853447MaRDI QIDQ3592089

Gérald Tenenbaum, Kevin Ford

Publication date: 12 September 2007

Published in: The Quarterly Journal of Mathematics (Search for Journal in Brave)

Abstract: Let H(x,y,z) be the number of integers $lex$ with a divisor in (y,z] and let H_1(x,y,z) be the number of integers $lex$ 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.

Full work available at URL: https://arxiv.org/abs/math/0607460

zbMATH Keywords

asymptotic formuladivisorsshort intervalsexactly one

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Distribution of integers with specified multiplicative constraints (11N25)

Related Items (4)

The distribution of integers with a divisor in a given interval ⋮ Unnamed Item ⋮ Distribution of rational numbers in short intervals ⋮ On the proximity of divisors of integers

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