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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.

Use the convention that a z-sieved number is a positive integer with no prime factor strictly below $z$; $1$ is included. Such integers are also called rough numbers. Using instead exclusion of primes at most $z$ changes an endpoint convention, not the following fixed-$u$ asymptotic.

The Buchstab function is the continuous function $w:[1,\infty )\to \mathbb{R}$ determined bySuccessive integration over intervals of length one determines the Buchstab function uniquely from its initial values.

A standard fixed-$u$ form of Buchstab theorem states that, for each fixed $u>1$, as $z\to \infty$,For $1<u\le 2$, this agrees with the Prime number theorem, since apart from an immaterial endpoint the z-sieved numbers in this range are $1$ and the primes from $z$ to $z^u$. The restriction $u>1$ matters: at $u=1$ the count is bounded, so the displayed asymptotic does not hold there.