A link between error terms when counting smooth and rough numbers

We establish a relationship between error terms appearing in estimates for the counting functions of smooth and rough numbers. We then apply this link to obtain an explicit upper bound for the error term in de Bruijn’s approximation $\mathrm{\Lambda }$ for the count of smooth numbers, from an explicit upper bound, due to Fan, for the error term in a variant of de Bruijn’s estimate for the count of rough numbers.