On integers \(\leq x\) having more than \(\log x\) divisors. (Sur les entiers inférieurs à \(x\) ayant plus de \(\log(x)\) diviseurs.) (French) Zbl 0839.11041

The authors are interested in the asymptotic behavior of the number of integers \(\leq x\) with more than \(\log (x)\) prime factors and, more generally, the behavior of the function \[ S_\lambda (x) = \# \bigl\{ n \leq x : \tau (n) \geq (\log x)^{\lambda \log 2} \bigr\}, \] where \(\tau (n)\) is the divisor function and \(\lambda\) a positive real number. In a recent paper [J. Number Theory 40, 146–164 (1992; Zbl 0745.11041)] M. Balazard, J.-L. Nicolas, C. Pomerance, and G. Tenenbaum showed that for fixed \(\lambda \geq 1\), \(S_\lambda (x)\) has order of magnitude \[ f (\lambda, x) = {x \over (\log x)^{\lambda \log \lambda - \lambda + 1} \sqrt {\log \log x}}; \] more precisely, the behavior of \(S_\lambda (x)\) is given by \[ S_\lambda (x) = c (\lambda) K(\lambda \log \log x) f(\lambda, x) \left( 1 + O \left( {1 \over \log \log x} \right) \right), \tag \(*\) \] where \(c (\lambda)\) is a positive constant and the function \(K (\theta)\) is bounded from above and below, left-continuous, and periodic with period 1.
In the paper under review, the authors investigate the behavior of the function \(K (\theta)\) in \((*)\). Their main result shows that in the case \(1 \leq \lambda \leq 2\) \[ \inf_{\theta} K (\theta) = K(0+), \quad \sup_\theta K (\theta) = K(0). \] Combining this result with numerical computations, they obtain numerical values for the lower and upper limits for the ratio \(S_\lambda (x)/f (\lambda, x) \), as \(x \to \infty\). For example, in the case \(\lambda = 1/ \log 2\) these limits are \(0.93827 \dots\) and \(1.148126 \dots\), respectively.


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


Zbl 0745.11041
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