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

DOI : https://doi.org/10.5802/jtnb.118

@article{JTNB_1994__6_2_327_0,
     author = {Del\'eglise, Marc and Nicolas, Jean-Louis},
     title = {Sur les entiers inf\'erieurs \`a $x$ ayant plus de $\log (x)$ diviseurs},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {327--357},
     publisher = {Universit\'e Bordeaux I},
     volume = {6},
     number = {2},
     year = {1994},
     doi = {10.5802/jtnb.118},
     zbl = {0839.11041},
     mrnumber = {1360649},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.118/}
}

Marc Deléglise; Jean-Louis Nicolas. Sur les entiers inférieurs à $x$ ayant plus de $\log (x)$ diviseurs. Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 327-357. doi : 10.5802/jtnb.118. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.118/