The distribution of integers with a divisor in a given interval

• Autores: Kevin Ford
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 168, Nº 2, 2008, págs. 367-433
• Idioma: inglés
• DOI: 10.4007/annals.2008.168.367
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• Resumen

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