In [Er79] asks the stronger version with $2$ replaced by any constant $c>1$.

The answer is yes (also to this stronger version), proved by Maier and Tenenbaum [MaTe84]. (Tenenbaum has told me that they received \$650 for their solution.)

In [Er64h] claimed a proof that the set of integers $n$ with divisors\[d_1<d_2<d_1(1+(\log n)^{-\beta})\]has density $1$ if $\beta<\log 3-1$, but this claim was retracted in [ErHa79]. Erdős and Hall [ErHa79] proved that this set has density $0$ if $\beta >\log 3-1$ (in a stronger quantitative form). The proof of Maier and Tenenbaum [MaTe84] proves that the density is $1$ if $\beta<\log 3-1$.

This is discussed in problem E3 of Guy's collection [Gu04].

See also [449] and [884].

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This page was last edited 29 December 2025.