A new relationship between error terms in smooth and rough number counting functions yields an explicit upper bound for the error in de Bruijn's smooth-number approximation using Fan's rough-number bound.
Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Third Edition, Graduate Studies in Mathematics, Vol
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A link between error terms when counting smooth and rough numbers
A new relationship between error terms in smooth and rough number counting functions yields an explicit upper bound for the error in de Bruijn's smooth-number approximation using Fan's rough-number bound.