On the density of certain sets of multiples. III (Q1355086)

In Parts I and II [Acta Arith. 69, 121-152 and 171-188 (1995; Zbl 0821.11045)] the author considered the set \(B_\lambda (n)\) of integers \(m\) for which we can find a \(d \mid n\) and \(d' \mid m\) such that \(d<d'\leq ( 1 + ( \log n )^{-\lambda } ) d \), and gave asymptotic formulas, valid for almost all values of \(n\), for the density of this set. Here the quantity \(N(x, \lambda , \varepsilon )\) of those values of \(n\leq x\) is estimated for which this density is \(<1-\varepsilon \). It is proved that for \(\lambda < \log 4 -1 \) we have \[ x ( \log x )^c \leq N(x, \lambda , \varepsilon ) \leq x \exp - c' \sqrt { \log \log x} \] with certain positive constants \(c,c'\) depending on \(\lambda \).