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The Quarterly Journal of Mathematics Advance Access originally published online on May 13, 2007
The Quarterly Journal of Mathematics 2007 58(2):187-201; doi:10.1093/qmath/ham001
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

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

Kevin Ford1,{dagger} and Gérald Tenenbaum2 {ddagger}

1 Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green St., Urbana, IL, 61801, USA
2 Institut Élie Cartan, Université Henri–Poincareé Nancy 1, B.P. 239, 54506 Vandœuvre-1ès-Nancy Cedex, France

{dagger} Corresponding author. E-mail: ford{at}math.uiuc.edu

Received 13 June 2006; revised 1 November 2006
  Abstract

We estimate the density of integers which have more than one divisor in an interval (y, z] with z {approx} y + y/(log y)log 4 – 1. As a consequence, we determine the precise range of z such that most integers which have at least one divisor in (y, z] have exactly one such divisor.

{ddagger} E-mail: gerald.tenenbaum{at}iecn.u-nancy.fr


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