A link between error terms when counting smooth and rough numbers

TL;DR

This paper uncovers a connection between the error terms in counting smooth and rough numbers, enabling explicit bounds for their estimation errors based on previous results.

Contribution

It introduces a novel link between the error terms in smooth and rough number counting functions and derives explicit bounds for these errors.

Findings

Established a relationship between error terms in smooth and rough number counts.

Derived explicit upper bounds for the error term in de Bruijn's approximation for smooth numbers.

Connected bounds for rough numbers to those for smooth numbers using the established link.

Abstract

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 $\Lambda$ 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.