Besicovitch [Be34] proved that $\liminf \delta(n)=0$. Erdős [Er35] proved that $\delta(n)=o(1)$. Erdős [Er60] proved that $\delta(n)=(\log n)^{-\alpha+o(1)}$ where\[\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.\]This estimate was refined by Tenenbaum [Te84], and the true growth rate of $\delta(n)$ was determined by Ford [Fo08] who proved\[\delta(n)\asymp \frac{1}{(\log n)^\alpha(\log\log n)^{3/2}}.\]Erdős asked this at Oberwolfach in 1986, and wrote he was 'quite sure' that $\delta_1(n)=o(\delta(n))$, but that 'recent results of Tenenbaum throw some doubt on this'.

Indeed, this was disproved by Ford [Fo08], who showed more generally that if $\delta_r(n)$ is the density of integers with exactly $r$ divisors in $(n,2n)$ then $\delta_r(n)\gg_r\delta(n)$.

See also [448], [692], and [693].

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This page was last edited 04 November 2025.