Sur les entiers inférieurs à $x$ ayant plus de $log\left(x\right)$ diviseurs

Deléglise, Marc  ; Nicolas, Jean-Louis 

Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 327-357

Résumé

Let $\tau \left(n\right)$ be the number of divisors of $n$ ; let us define

$$S_{\lambda }\left(x\right)=\left\{\begin{array}{cc}\mathrm{Card}\left\{n\le x;\tau \left(n\right)\ge {(logx)}^{\lambda log2}\right\}\hfill & \text{if}\lambda \ge 1\hfill \\ \mathrm{Card}\left\{n\le x;\tau \left(n\right)<{(logx)}^{\lambda log2}\right\}\hfill & \text{if}\lambda <1\hfill \end{array}\right.$$

It has been shown that, if we set

$$f(\lambda ,x)=\frac{x}{{(logx)}^{\lambda log\lambda -\lambda +1}\sqrt{loglogx}}$$

the quotient $S\lambda \left(x\right)/f(\lambda ,x)$ is bounded for $\lambda$ fixed. The aim of this paper is to give an explicit value for the inferior and superior limits of this quotient when $\lambda \ge 2$. For instance, when $\lambda =1/log2$, we prove

$$liminf\frac{S_{\lambda }\left(x\right)}{f(\lambda ,x)}=0.938278681143\cdots$$

and

$$liminf\frac{S_{\lambda }\left(x\right)}{f(\lambda ,x)}=1.148126773469\cdots$$