I am currently reading section 5 of Pomerance's paper The Number Field Sieve and I have a few questions about smooth numbers.

A number $x\in\mathbb Z_{\ge1}$ is called $z$-smooth if every prime divisor $p$ of $x$ satisfies $p\le z$. So if we want to check if a given natural number is $z$-smooth, we could simply trial divide by all prime numbers $p\le z$, which should take about $\pi(z)$ steps ($\pi$ = prime-counting function).

However, Pomerance states that it would take about $z$ steps, which I think should be $\pi(z)$ (as pointed out above). And he also states that using his *early abort strategy* one would need about $z^{1/2}$ steps, which I think should read $\pi(z)^{1/2}$.

I guess that I am missing his point, since he is a very accomplished number theorist, but I am not sure.