Sur les entiers inférieurs à $x$ ayant plus de $log\left(x\right)$ diviseurs
Journal de Théorie des Nombres de Bordeaux, Volume 6 (1994) no. 2, pp. 327-357.

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 {\left(logx\right)}^{\lambda log2}\right\}\hfill & \text{if}\phantom{\rule{4pt}{0ex}}\lambda \ge 1\hfill \\ \mathrm{Card}\left\{n\le x;\tau \left(n\right)<{\left(logx\right)}^{\lambda log2}\right\}\hfill & \text{if}\phantom{\rule{4pt}{0ex}}\lambda <1\hfill \end{array}\right\$ It has been shown that, if we set $f\left(\lambda ,x\right)=\frac{x}{{\left(logx\right)}^{\lambda log\lambda -\lambda +1}\sqrt{loglogx}}$ the quotient $S\lambda \left(x\right)/f\left(\lambda ,x\right)$ 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 $liminf\frac{{S}_{\lambda }\left(x\right)}{f\left(\lambda ,x\right)}=0.938278681143\cdots$ and $liminf\frac{{S}_{\lambda }\left(x\right)}{f\left(\lambda ,x\right)}=1.148126773469\cdots$

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author = {Marc Del\'eglise and Jean-Louis Nicolas},
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},
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year = {1994},
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language = {fr},
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Marc Deléglise; Jean-Louis Nicolas. Sur les entiers inférieurs à $x$ ayant plus de $\log (x)$ diviseurs. Journal de Théorie des Nombres de Bordeaux, Volume 6 (1994) no. 2, pp. 327-357. doi : 10.5802/jtnb.118. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.118/`

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