A $n$-smooth (rational) integer is a positive integer with prime factors all less than or equal to $n$. We can define $\Psi (a,n)$ as the number of $n$-smooth integers less than or equal to $a$. It is known that for a fixed $n$,
$\Psi (x,n) \sim 1/(\pi(x)!)\Pi_{p\leq x} \log(x)/\log(p)$,
where $p$ denotes a prime. If we let the second argument vary as well, then we have
$\Psi (x,y) = x\cdot \rho(\log(x)/\log(y))+O(x/\log(y)),$
with $\rho$ is Dickman's function.
Now let $K/\mathbb{Q}$ be an algebraic number field. We can define an $n$-smooth ideal in $\mathcal{O}_K$ as an ideal that is divisible entirely by prime ideals with norm less than or equal to $n$.
Question: Are there known results/bounds on $\Psi(x,y)$ in $K$, where $\Psi(x,y)$ is the number of $y$-smooth ideals in $\mathcal{O}_K$ with norm less than or equal to $x$?