## 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.

### MSC:

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

Zbl 0745.11041
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### References:

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