Let $\xi$ be a positive parameter. We say a positive integer $n$ is $\xi$-smooth, or friable, if for all primes $p$ dividing $n$, we have $p \leq \xi$. Let $T(X, \xi)$ denote the set of $\xi$-smooth numbers up to $X$, and $\Psi(X,\xi) = \# T(X,\xi)$. Then it is known that for all $a > 0$, we have $\Psi(X,X^{1/a}) \sim \rho(a) X$, where $\rho$ is Dickman's function.

Let $f(x) \in \mathbb{Z}[x]$ be of degree $d$ at least $3$, irreducible, and have no fixed prime divisor. Define $$\displaystyle \mathcal{S}_f(B) = \{n \in \mathbb{N} : n \leq B, \text{there exists } t \in \mathbb{N} \text{ s.t. } f(t) = n\}.$$ It is clear that $\# \mathcal{S}_f(B) \sim B^{1/d}$ as $B \rightarrow \infty$. However, it is usually difficult to obtain finer arithmetic properties for $\mathcal{S}_f(B)$. For instance, it is not known whether $\mathcal{S}_f(B)$ contains infinitely many primes when $\deg f > 1$.

What about smooth numbers? That is, can we confirm that

$$\displaystyle \# (T(B, B^{\kappa(d)}) \cap \mathcal{S}_f(B)) \ll \rho(1/\kappa(d)) \# \mathcal{S}_f(B)$$

for, say, $\kappa(d) = 1/(d-1)$?


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