An Erdős–Kac theorem for integers with dense divisors

Author:

Tenenbaum Gérald12,Weingartner Andreas12ORCID

Affiliation:

1. Institut Élie Cartan, Université de Lorraine , B.P. 70239, F–54506 Vandœuvre-lès-Nancy Cedex, France

2. Department of Mathematics, Southern Utah University , 351 West University Boulevard, Cedar City, UT 84720, USA

Abstract

Abstract We show that for large integers n, whose ratios of consecutive divisors are bound above by an arbitrary constant, the number of prime factors follows an approximate normal distribution, with mean $C \log_2 n$ and variance $V \log_2 n$, where $C=1/(1-{\rm e}^{-\gamma})\approx 2.280$ and V ≈ 0.414. This result is then generalized in two different directions.

Publisher

Oxford University Press (OUP)

Reference20 articles.

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4. On a class of differential-difference equations arising in number theory;Hildebrand;J. Anal. Math.,1993

5. Natural divisors and the Brownian motion;Manstavičius;J. Théor. Nombres Bordeaux,1996

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