 Buchstab function

The continuous Buchstab function satisfies $w(u)=1/u$ for $1\le u\le 2$ and $\frac{d}{du}(uw(u))=w(u-1)$ for $u>2$. It describes the density of rough numbers.

The Buchstab function is a special arithmetic function used in number theory, particularly in the study of prime numbers and their distribution. It is often denoted by \( B(n) \) or \( b(n) \) and is related to the behavior of the prime counting function and the distribution of prime numbers in relation to composite numbers.