= Solution
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 by
$$
\boxed{w(u)=\frac1u\quad(1\le u\le2),\qquad\frac{d}{du}(u w(u))=w(u-1)\quad(u>2).}
$$
Successive 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 $z\to\infty$,
$$
\boxed{\Phi(z^u,z)\sim\frac{z^u}{\log z}\,w(u).}
$$
For $1<u\le2$, 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.
Back to article page
